Terminados apuntes de Álgebra Conmutativa

8 files changed, 12077 insertions, 4419 deletions

diff --git a/ac/n.lyx b/ac/n.lyx
index 5261b22..38e61b7 100644
--- a/ac/n.lyx
+++ b/ac/n.lyx
@@ -153,6 +153,31 @@ Alberto del Valle Robles.
 Clases de Manuel Saorín Castaño.
 \end_layout
 
+\begin_layout Itemize
+Manuel Saorín Castaño.
+ 
+\emph on
+Capítulo IV: Módulos sobre dominios de ideales principales
+\emph default
+.
+\end_layout
+
+\begin_layout Itemize
+
+\lang english
+Donald Knuth.
+ 
+\emph on
+The Art of Computer Programming.
+ Volume 1: Fundamental Algorithms
+\emph default
+\lang spanish
+, 3rd.
+ ed.
+ (1997), pp.
+ 45–87.
+\end_layout
+
 \begin_layout Standard
 \begin_inset ERT
 status open
@@ -238,5 +263,62 @@ filename "n4.lyx"
 
 \end_layout
 
+\begin_layout Chapter
+Endomorfismos vectoriales en dimensión finita
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n5.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+\start_of_appendix
+Grupos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "na.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Anillos de polinomios
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "nb.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Coeficientes binomiales
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "nc.lyx"
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document
diff --git a/ac/n1.lyx b/ac/n1.lyx
index 2cd5d09..16e7e5f 100644
--- a/ac/n1.lyx
+++ b/ac/n1.lyx
@@ -519,7 +519,7 @@ status open
 \end_inset
 
 , definimos 
-\begin_inset Formula $0_{\mathbb{Z}}a\coloneqq 0$
+\begin_inset Formula $0_{\mathbb{Z}}a\coloneqq0$
 \end_inset
 
 , y para 
@@ -527,16 +527,16 @@ status open
 \end_inset
 
 , 
-\begin_inset Formula $na\coloneqq (n-1)a+a$
+\begin_inset Formula $na\coloneqq(n-1)a+a$
 \end_inset
 
  y 
-\begin_inset Formula $(-n)a\coloneqq -(na)$
+\begin_inset Formula $(-n)a\coloneqq-(na)$
 \end_inset
 
 .
  Definimos 
-\begin_inset Formula $a^{0_{\mathbb{Z}}}\coloneqq 1_{A}$
+\begin_inset Formula $a^{0_{\mathbb{Z}}}\coloneqq1_{A}$
 \end_inset
 
 , para 
@@ -552,7 +552,7 @@ status open
 \end_inset
 
  es invertible, 
-\begin_inset Formula $a^{-n}\coloneqq (a^{-1})^{n}$
+\begin_inset Formula $a^{-n}\coloneqq(a^{-1})^{n}$
 \end_inset
 
 .
@@ -2431,11 +2431,7 @@ Si
 \end_layout
 
 \begin_layout Standard
-[...]
-\end_layout
-
-\begin_layout Standard
-Dado un dominio 
+[...] Dado un dominio 
 \begin_inset Formula $D$
 \end_inset
 
@@ -2493,7 +2489,7 @@ equivalentes
 \end_inset
 
  de 
-\begin_inset Formula $\mathbb{N}_{n}\coloneqq \{1,\dots,n\}$
+\begin_inset Formula $\mathbb{N}_{n}\coloneqq\{1,\dots,n\}$
 \end_inset
 
  tal que para 
@@ -3284,7 +3280,7 @@ subanillo primo
 \end_inset
 
  a 
-\begin_inset Formula $\mathbb{Z}1\coloneqq \{n1_{A}\}_{n\in\mathbb{Z}}$
+\begin_inset Formula $\mathbb{Z}1\coloneqq\{n1_{A}\}_{n\in\mathbb{Z}}$
 \end_inset
 
 , el menor subanillo de 
@@ -4248,7 +4244,7 @@ Dado
 \end_inset
 
 , llamamos 
-\begin_inset Formula $\mathbb{Z}_{n}\coloneqq \frac{\mathbb{Z}}{n\mathbb{Z}}=\{0+n\mathbb{Z},\dots,(n-1)+n\mathbb{Z}\}$
+\begin_inset Formula $\mathbb{Z}_{n}\coloneqq\frac{\mathbb{Z}}{n\mathbb{Z}}=\{0+n\mathbb{Z},\dots,(n-1)+n\mathbb{Z}\}$
 \end_inset
 
 .
@@ -8655,2756 +8651,5 @@ end{exinfo}
 
 \end_layout
 
-\begin_layout Section
-Dominios euclídeos
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{reminder}{GyA}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Dado un dominio 
-\begin_inset Formula $D\neq0$
-\end_inset
-
-, una función 
-\begin_inset Formula $\delta:D\setminus\{0\}\to\mathbb{N}$
-\end_inset
-
- es 
-\series bold
-euclídea
-\series default
- si cumple:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\forall a,b\in D\setminus\{0\},(a\mid b\implies\delta(a)\leq\delta(b))$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D\mid(a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Un 
-\series bold
-dominio euclídeo
-\series default
- es uno que admite una función euclídea.
-\end_layout
-
-\begin_layout Enumerate
-El valor absoluto es una función euclídea en 
-\begin_inset Formula $\mathbb{Z}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-El cuadrado del módulo complejo es una función euclídea en 
-\begin_inset Formula $\mathbb{Z}[i]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $\delta$
-\end_inset
-
- una función euclídea en 
-\begin_inset Formula $D$
-\end_inset
-
-, 
-\begin_inset Formula $I$
-\end_inset
-
- un ideal de 
-\begin_inset Formula $D$
-\end_inset
-
- y 
-\begin_inset Formula $a\in I\setminus\{0\}$
-\end_inset
-
-, entonces 
-\begin_inset Formula 
-\[
-I=(a)\iff\forall x\in I\setminus\{0\},\delta(a)\leq\delta(x).
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-[...] Todo dominio euclídeo es DIP.
- Si 
-\begin_inset Formula $\delta$
-\end_inset
-
- es una función euclídea en 
-\begin_inset Formula $D$
-\end_inset
-
-, un elemento 
-\begin_inset Formula $a\in D$
-\end_inset
-
- es una unidad si y sólo si 
-\begin_inset Formula $\delta(a)=\delta(1)$
-\end_inset
-
-, si y sólo si 
-\begin_inset Formula $\forall x\in D\setminus\{0\},\delta(a)\leq\delta(x)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{reminder}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Cuerpos de fracciones
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{reminder}{GyA}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $D\neq0$
-\end_inset
-
- un dominio y 
-\begin_inset Formula $X\coloneqq D\times(D\setminus\{0\})$
-\end_inset
-
-, definimos la relación binaria 
-\begin_inset Formula 
-\[
-(a_{1},s_{1})\sim(a_{2},s_{2}):\iff a_{1}s_{2}=a_{2}s_{1}.
-\]
-
-\end_inset
-
- Esta relación es de equivalencia.
- Llamamos 
-\begin_inset Formula $a/s\coloneqq \frac{a}{s}\coloneqq [(a,s)]\in Q(D)\coloneqq X/\sim$
-\end_inset
-
-, y las operaciones
-\begin_inset Formula 
-\begin{align*}
-\frac{a_{1}}{s_{1}}+\frac{a_{2}}{s_{2}} & :=\frac{a_{1}s_{2}+a_{2}s_{1}}{s_{1}s_{2}}, & \frac{a_{1}}{s_{1}}\cdot\frac{a_{2}}{s_{2}} & :=\frac{a_{1}a_{2}}{s_{1}s_{2}},
-\end{align*}
-
-\end_inset
-
-están bien definidas.
-\end_layout
-
-\begin_layout Standard
-Para 
-\begin_inset Formula $a,b\in D$
-\end_inset
-
- y 
-\begin_inset Formula $s,t\in D\setminus\{0\}$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\frac{a}{s}=\frac{0}{1}\iff a=0$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\frac{a}{s}=\frac{1}{1}\iff a=s$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\frac{at}{st}=\frac{a}{s}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\frac{a}{s}=\frac{b}{s}\iff a=b$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\frac{a}{s}+\frac{b}{s}=\frac{a+b}{s}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-[...] 
-\begin_inset Formula $(Q(D),+,\cdot)$
-\end_inset
-
- es un cuerpo llamado 
-\series bold
-cuerpo de fracciones
-\series default
- o 
-\series bold
-de cocientes
-\series default
- de 
-\begin_inset Formula $D$
-\end_inset
-
- cuyo cero es 
-\begin_inset Formula $\frac{0}{1}$
-\end_inset
-
- y cuyo uno es 
-\begin_inset Formula $\frac{1}{1}$
-\end_inset
-
- .
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $\mathbb{Q}$
-\end_inset
-
- es el cuerpo de fracciones de 
-\begin_inset Formula $\mathbb{Z}$
-\end_inset
-
-.
- [...] 
-\begin_inset Formula $u:D\to Q(D)$
-\end_inset
-
- dada por 
-\begin_inset Formula $u(a)\coloneqq a/1$
-\end_inset
-
- es un homomorfismo inyectivo, por lo que podemos ver a 
-\begin_inset Formula $D$
-\end_inset
-
- como un subdominio de 
-\begin_inset Formula $Q(D)$
-\end_inset
-
- identificando a cada 
-\begin_inset Formula $a\in D$
-\end_inset
-
- con 
-\begin_inset Formula $a/1\in Q(D)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{samepage}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Propiedad universal del cuerpo de fracciones:
-\series default
- Dados un dominio 
-\begin_inset Formula $D$
-\end_inset
-
- y 
-\begin_inset Formula $u:D\to Q(D)$
-\end_inset
-
- dada por 
-\begin_inset Formula $u(a)\coloneqq a/1$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $K$
-\end_inset
-
- un cuerpo y 
-\begin_inset Formula $f:D\to K$
-\end_inset
-
- un homomorfismo inyectivo, el único homomorfismo de cuerpos 
-\begin_inset Formula $\tilde{f}:Q(D)\to K$
-\end_inset
-
- con 
-\begin_inset Formula $\tilde{f}\circ u=f$
-\end_inset
-
- viene dado por 
-\begin_inset Formula $\tilde{f}(\frac{a}{s})=f(a)f(s)^{-1}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $K$
-\end_inset
-
- un cuerpo no trivial y 
-\begin_inset Formula $g,h:Q(D)\to K$
-\end_inset
-
- homomorfismos que coinciden en 
-\begin_inset Formula $D$
-\end_inset
-
-, entonces 
-\begin_inset Formula $g=h$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $F$
-\end_inset
-
- un cuerpo no trivial y 
-\begin_inset Formula $v:D\to F$
-\end_inset
-
- un homomorfismo inyectivo tal que para todo cuerpo 
-\begin_inset Formula $K$
-\end_inset
-
- y homomorfismo inyectivo 
-\begin_inset Formula $f:D\to K$
-\end_inset
-
- existe un único homomorfismo 
-\begin_inset Formula $\tilde{f}:F\to K$
-\end_inset
-
- con 
-\begin_inset Formula $\tilde{f}\circ v=f$
-\end_inset
-
-, entonces existe un isomorfismo 
-\begin_inset Formula $\phi:F\to Q(D)$
-\end_inset
-
- con 
-\begin_inset Formula $\phi\circ v=u$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{samepage}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $D$
-\end_inset
-
- un dominio, 
-\begin_inset Formula $K$
-\end_inset
-
- un cuerpo no trivial y 
-\begin_inset Formula $f:D\to K$
-\end_inset
-
- un homomorfismo inyectivo, 
-\begin_inset Formula $K$
-\end_inset
-
- contiene un subcuerpo isomorfo a 
-\begin_inset Formula $Q(D)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-De aquí, para 
-\begin_inset Formula $m\in\mathbb{Z}$
-\end_inset
-
-, 
-\begin_inset Formula $Q(\mathbb{Z}[\sqrt{m}])\cong\mathbb{Q}[\sqrt{m}]$
-\end_inset
-
-, lo que nos permite identificar los elementos de 
-\begin_inset Formula $Q(\mathbb{Z}[\sqrt{m}])$
-\end_inset
-
- con los de 
-\begin_inset Formula $\mathbb{Q}[\sqrt{m}]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Sea 
-\begin_inset Formula $K$
-\end_inset
-
- un cuerpo no trivial, existe un subcuerpo 
-\begin_inset Formula $K'$
-\end_inset
-
- de 
-\begin_inset Formula $K$
-\end_inset
-
- llamado 
-\series bold
-subcuerpo primo
-\series default
- de 
-\begin_inset Formula $K$
-\end_inset
-
- contenido en cualquier subcuerpo de 
-\begin_inset Formula $K$
-\end_inset
-
-, y este es isomorfo a 
-\begin_inset Formula $\mathbb{Z}_{p}$
-\end_inset
-
- si la característica de 
-\begin_inset Formula $K$
-\end_inset
-
- es un entero primo 
-\begin_inset Formula $p$
-\end_inset
-
- o a 
-\begin_inset Formula $\mathbb{Q}$
-\end_inset
-
- en caso contrario.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{reminder}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Polinomios
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{reminder}{GyA}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $A$
-\end_inset
-
- es un subanillo de 
-\begin_inset Formula $A[X]$
-\end_inset
-
- identificando los elementos de 
-\begin_inset Formula $A$
-\end_inset
-
- con los 
-\series bold
-polinomios constantes
-\series default
-, de la forma 
-\begin_inset Formula $P(X)=a_{0}$
-\end_inset
-
-.
- Dado un ideal 
-\begin_inset Formula $I$
-\end_inset
-
- de 
-\begin_inset Formula $A$
-\end_inset
-
-, 
-\begin_inset Formula $\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0}\in I\}$
-\end_inset
-
- e 
-\begin_inset Formula $I[X]\coloneqq \{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0},\dots,a_{n}\in I\}$
-\end_inset
-
- son ideales de 
-\begin_inset Formula $A[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Dado 
-\begin_inset Formula $p\coloneqq \sum_{k\in\mathbb{N}}p_{k}X^{k}\in A[X]\setminus\{0\}$
-\end_inset
-
-, llamamos 
-\series bold
-grado
-\series default
- de 
-\begin_inset Formula $p$
-\end_inset
-
- a 
-\begin_inset Formula $\text{gr}(p)\coloneqq \max\{k\in\mathbb{N}\mid p_{k}\neq0\}$
-\end_inset
-
-, 
-\series bold
-coeficiente
-\series default
- de 
-\series bold
-grado
-\series default
- 
-\begin_inset Formula $k$
-\end_inset
-
- de 
-\begin_inset Formula $p$
-\end_inset
-
- a 
-\begin_inset Formula $p_{k}$
-\end_inset
-
-, 
-\series bold
-coeficiente independiente
-\series default
- al de grado 0 y 
-\series bold
-coeficiente principal
-\series default
- al de grado 
-\begin_inset Formula $\text{gr}(p)$
-\end_inset
-
-.
- Un polinomio es 
-\series bold
-mónico
-\series default
- si su coeficiente principal es 1.
- El polinomio 0 tiene grado 
-\begin_inset Formula $-\infty$
-\end_inset
-
- por convención.
-\end_layout
-
-\begin_layout Standard
-Un 
-\series bold
-monomio
-\series default
- es un polinomio de la forma 
-\begin_inset Formula $aX^{n}$
-\end_inset
-
- con 
-\begin_inset Formula $a\in A$
-\end_inset
-
- y 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
-.
- Todo polinomio en 
-\begin_inset Formula $A[X]$
-\end_inset
-
- se escribe como suma finita de monomios de distinto grado de forma única
- salvo orden.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $P,Q\in A[X]\setminus\{0\}$
-\end_inset
-
- tienen coeficientes principales respectivos 
-\begin_inset Formula $p$
-\end_inset
-
- y 
-\begin_inset Formula $q$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\text{gr}(P+Q)\leq\max\{\text{gr}(P),\text{gr}(Q)\}$
-\end_inset
-
-, con desigualdad estricta si y sólo si 
-\begin_inset Formula $\text{gr}(P)=\text{gr}(Q)$
-\end_inset
-
- y 
-\begin_inset Formula $p+q=0$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\text{gr}(PQ)\leq\text{gr}(P)+\text{gr}(Q)$
-\end_inset
-
-, con igualdad si y sólo si 
-\begin_inset Formula $pq\neq0$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $A[X]$
-\end_inset
-
- no es un cuerpo.
- Es un dominio si y sólo si lo es 
-\begin_inset Formula $A$
-\end_inset
-
-, en cuyo caso llamamos 
-\series bold
-cuerpo de las funciones racionales
-\series default
- sobre 
-\begin_inset Formula $A$
-\end_inset
-
- al cuerpo de fracciones de 
-\begin_inset Formula $A[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-[...] 
-\series bold
-Propiedad universal del anillo de polinomios
-\series default
- (
-\series bold
-PUAP
-\series default
-)
-\series bold
-:
-\series default
- Sean 
-\begin_inset Formula $A$
-\end_inset
-
- un anillo y 
-\begin_inset Formula $u:A\to A[X]$
-\end_inset
-
- el homomorfismo inclusión:
-\end_layout
-
-\begin_layout Enumerate
-Para cada homomorfismo de anillos conmutativos 
-\begin_inset Formula $f:A\to B$
-\end_inset
-
- y 
-\begin_inset Formula $b\in B$
-\end_inset
-
-, el único homomorfismo 
-\begin_inset Formula $\tilde{f}:A[X]\to B$
-\end_inset
-
- tal que 
-\begin_inset Formula $\tilde{f}(X)=b$
-\end_inset
-
- y 
-\begin_inset Formula $\tilde{f}\circ u=f$
-\end_inset
-
- es
-\begin_inset Formula 
-\[
-\tilde{f}\left(\sum_{n}p_{n}X^{n}\right):=\sum_{n}f(p_{n})b^{n}.
-\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $A[X]$
-\end_inset
-
- y 
-\begin_inset Formula $u$
-\end_inset
-
- están determinados salvo isomorfismos por la propiedad universal: dados
- un homomorfismo de anillos 
-\begin_inset Formula $v:A\to P$
-\end_inset
-
- y 
-\begin_inset Formula $t\in P$
-\end_inset
-
- tales que, para cada homomorfismo de anillos 
-\begin_inset Formula $f:A\to B$
-\end_inset
-
- y 
-\begin_inset Formula $b\in B$
-\end_inset
-
-, existe un único 
-\begin_inset Formula $\tilde{f}:P\to B$
-\end_inset
-
- tal que 
-\begin_inset Formula $\tilde{f}\circ v=f$
-\end_inset
-
- y 
-\begin_inset Formula $\tilde{f}(t)=b$
-\end_inset
-
-, existe un isomorfismo 
-\begin_inset Formula $\phi:A[X]\to P$
-\end_inset
-
- tal que 
-\begin_inset Formula $\phi\circ u=v$
-\end_inset
-
- y 
-\begin_inset Formula $\phi(X)=t$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Así:
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $A$
-\end_inset
-
- es un subanillo de 
-\begin_inset Formula $B$
-\end_inset
-
- y 
-\begin_inset Formula $b\in B$
-\end_inset
-
-, el 
-\series bold
-homomorfismo de sustitución
-\series default
- o 
-\series bold
-de evaluación
-\series default
- en 
-\begin_inset Formula $b$
-\end_inset
-
- es 
-\begin_inset Formula $S_{b}:A[X]\to B$
-\end_inset
-
- dado por
-\begin_inset Formula 
-\[
-S_{b}(p):=p(b):=\sum_{n}p_{n}b^{n},
-\]
-
-\end_inset
-
-y su imagen es el subanillo generado por 
-\begin_inset Formula $A\cup\{b\}$
-\end_inset
-
-, llamado 
-\begin_inset Formula $A[b]$
-\end_inset
-
-.
- Todo 
-\begin_inset Formula $p\in A[X]$
-\end_inset
-
- induce una 
-\series bold
-función polinómica
-\series default
- 
-\begin_inset Formula $\hat{p}:B\to B$
-\end_inset
-
- dada por 
-\begin_inset Formula $\hat{p}(b)\coloneqq S_{b}(p)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Dado 
-\begin_inset Formula $a\in A$
-\end_inset
-
-, el homomorfismo de sustitución 
-\begin_inset Formula $S_{X+a}$
-\end_inset
-
- es un automorfismo de 
-\begin_inset Formula $A[X]$
-\end_inset
-
- con inverso 
-\begin_inset Formula $S_{X-a}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $A$
-\end_inset
-
- es un anillo conmutativo, 
-\begin_inset Formula $\frac{A[X]}{(X)}\cong A$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Todo homomorfismo de anillos 
-\begin_inset Formula $f:A\to B$
-\end_inset
-
- induce un homomorfismo 
-\begin_inset Formula $\hat{f}:A[X]\to B[X]$
-\end_inset
-
- dado por
-\begin_inset Formula 
-\[
-\hat{f}(p)=\sum_{n}f(p_{n})X^{n},
-\]
-
-\end_inset
-
-que es inyectivo o suprayectivo si lo es 
-\begin_inset Formula $f$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $A$
-\end_inset
-
- es un subanillo de 
-\begin_inset Formula $B$
-\end_inset
-
-, 
-\begin_inset Formula $A[X]$
-\end_inset
-
- lo es de 
-\begin_inset Formula $B[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $I$
-\end_inset
-
- es un ideal de 
-\begin_inset Formula $A$
-\end_inset
-
-, el 
-\series bold
-homomorfismo de reducción de coeficientes módulo 
-\begin_inset Formula $I$
-\end_inset
-
-
-\series default
- es 
-\begin_inset Formula $\tilde{\pi}:A[X]\to(A/I)[X]$
-\end_inset
-
- dado por
-\begin_inset Formula 
-\[
-\tilde{\pi}(p):=\sum_{n}(p_{n}+I)X^{n}.
-\]
-
-\end_inset
-
-Su núcleo es 
-\begin_inset Formula $I[X]$
-\end_inset
-
-, por lo que 
-\begin_inset Formula $(A/I)[X]\cong\frac{A[X]}{I[X]}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-[...] Sean 
-\begin_inset Formula $f,g\in A[X]$
-\end_inset
-
-, si el coeficiente principal de 
-\begin_inset Formula $g$
-\end_inset
-
- es invertible en 
-\begin_inset Formula $A$
-\end_inset
-
-, existen dos únicos polinomios 
-\begin_inset Formula $q,r\in A[X]$
-\end_inset
-
-, llamados respectivamente 
-\series bold
-cociente
-\series default
- y 
-\series bold
-resto
-\series default
- de la 
-\series bold
-división
-\series default
- de 
-\begin_inset Formula $f$
-\end_inset
-
- entre 
-\begin_inset Formula $g$
-\end_inset
-
-, tales que 
-\begin_inset Formula $f=gq+r$
-\end_inset
-
- y 
-\begin_inset Formula $\text{gr}(r)<\text{gr}(g)$
-\end_inset
-
- [...].
- En particular, el grado es una función euclídea.
- 
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema del resto:
-\series default
- Dados 
-\begin_inset Formula $f\in A[X]$
-\end_inset
-
- y 
-\begin_inset Formula $a\in A$
-\end_inset
-
-, el resto de 
-\begin_inset Formula $f$
-\end_inset
-
- entre 
-\begin_inset Formula $X-a$
-\end_inset
-
- es 
-\begin_inset Formula $f(a)$
-\end_inset
-
-.
- De aquí se obtiene el 
-\series bold
-teorema de Ruffini
-\series default
-, que dice que 
-\begin_inset Formula $f$
-\end_inset
-
- es divisible por 
-\begin_inset Formula $X-a$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $f(a)=0$
-\end_inset
-
-, en cuyo caso 
-\begin_inset Formula $a$
-\end_inset
-
- es una 
-\series bold
-raíz
-\series default
- de 
-\begin_inset Formula $f$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Para 
-\begin_inset Formula $f\in A[X]\setminus\{0\}$
-\end_inset
-
- y 
-\begin_inset Formula $a\in A$
-\end_inset
-
-, existe 
-\begin_inset Formula $m\coloneqq \max\{k\in\mathbb{N}\mid(X-a)^{k}\mid f\}$
-\end_inset
-
-.
- Llamamos a 
-\begin_inset Formula $m$
-\end_inset
-
- 
-\series bold
-multiplicidad
-\series default
- de 
-\begin_inset Formula $a$
-\end_inset
-
- en 
-\begin_inset Formula $f$
-\end_inset
-
-, y 
-\begin_inset Formula $a$
-\end_inset
-
- es raíz de 
-\begin_inset Formula $f$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $m\geq1$
-\end_inset
-
-.
- Decimos que 
-\begin_inset Formula $a$
-\end_inset
-
- es una 
-\series bold
-raíz simple
-\series default
- de 
-\begin_inset Formula $f$
-\end_inset
-
- si 
-\begin_inset Formula $m=1$
-\end_inset
-
- y que es una 
-\series bold
-raíz compuesta
-\series default
- si 
-\begin_inset Formula $m>1$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-La multiplicidad de 
-\begin_inset Formula $a$
-\end_inset
-
- en 
-\begin_inset Formula $f$
-\end_inset
-
- es el único natural 
-\begin_inset Formula $m$
-\end_inset
-
- tal que 
-\begin_inset Formula $f=(X-a)^{m}g$
-\end_inset
-
- para algún 
-\begin_inset Formula $g\in A[X]$
-\end_inset
-
- del que 
-\begin_inset Formula $a$
-\end_inset
-
- no es raíz.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $D$
-\end_inset
-
- es un dominio, 
-\begin_inset Formula $f\in D[X]\setminus\{0\}$
-\end_inset
-
-, 
-\begin_inset Formula $a_{1},\dots,a_{n}$
-\end_inset
-
- son 
-\begin_inset Formula $n$
-\end_inset
-
- elementos de 
-\begin_inset Formula $D$
-\end_inset
-
- y 
-\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}\in\mathbb{Z}^{>0}$
-\end_inset
-
- con 
-\begin_inset Formula $(X-a_{k})^{\alpha_{k}}\mid f$
-\end_inset
-
- para cada 
-\begin_inset Formula $k$
-\end_inset
-
-, entonces 
-\begin_inset Formula $(X-a_{1})^{\alpha_{1}}\cdots(X-a_{n})^{\alpha_{n}}\mid f$
-\end_inset
-
-, por lo que 
-\begin_inset Formula $\sum_{k=1}^{n}\alpha_{k}\leq\text{gr}(f)$
-\end_inset
-
- y, en particular, la suma de las multiplicidades de las raíces de 
-\begin_inset Formula $f$
-\end_inset
-
-, y el número de raíces, no son superiores a 
-\begin_inset Formula $\text{gr}(f)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Principio de las identidades polinómicas:
-\series default
- Sea 
-\begin_inset Formula $D$
-\end_inset
-
- un dominio:
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $f,g\in D[X]$
-\end_inset
-
-, si las funciones polinómicas 
-\begin_inset Formula $f,g:D\to D$
-\end_inset
-
- coinciden en 
-\begin_inset Formula $m$
-\end_inset
-
- elementos de 
-\begin_inset Formula $D$
-\end_inset
-
- con 
-\begin_inset Formula $m>\text{gr}(f),\text{gr}(g)$
-\end_inset
-
-, los polinomios 
-\begin_inset Formula $f$
-\end_inset
-
- y 
-\begin_inset Formula $g$
-\end_inset
-
- son iguales.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $D$
-\end_inset
-
- es infinito si y sólo si cualquier par de polinomios distintos en 
-\begin_inset Formula $D[X]$
-\end_inset
-
- define dos funciones polinómicas distintas en 
-\begin_inset Formula $D$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Como ejemplo de lo anterior, por el teorema pequeño de Fermat, dado un primo
- 
-\begin_inset Formula $p$
-\end_inset
-
-, todos los elementos de 
-\begin_inset Formula $\mathbb{Z}_{p}$
-\end_inset
-
- son raíces de 0 y 
-\begin_inset Formula $X^{p}-X$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Dado un anillo conmutativo 
-\begin_inset Formula $A$
-\end_inset
-
-, definimos la 
-\series bold
-derivada
-\series default
- de 
-\begin_inset Formula $P\coloneqq \sum_{k}a_{k}X^{k}\in A[X]$
-\end_inset
-
- como 
-\begin_inset Formula $P'\coloneqq D(P)\coloneqq \sum_{k\geq1}ka_{k}X^{k-1}$
-\end_inset
-
-, y escribimos 
-\begin_inset Formula $P^{(0)}\coloneqq P$
-\end_inset
-
- y 
-\begin_inset Formula $P^{(n+1)}\coloneqq P^{(n)\prime}$
-\end_inset
-
-.
- Dados 
-\begin_inset Formula $a,b\in A$
-\end_inset
-
- y 
-\begin_inset Formula $P,Q\in A[X]$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $(aP+bQ)'=aP'+bQ'$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $(PQ)'=P'Q+PQ'$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $(P^{n})'=nP^{n-1}P'$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Dados un dominio 
-\begin_inset Formula $D$
-\end_inset
-
- de característica 0, 
-\begin_inset Formula $P\in D[X]\setminus\{0\}$
-\end_inset
-
- y 
-\begin_inset Formula $a\in D$
-\end_inset
-
-, la multiplicidad de 
-\begin_inset Formula $a$
-\end_inset
-
- en 
-\begin_inset Formula $P$
-\end_inset
-
- es el menor 
-\begin_inset Formula $m\in\mathbb{N}_{0}$
-\end_inset
-
- con 
-\begin_inset Formula $P^{(m)}(a)\neq0$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Standard
-Dado un anillo 
-\begin_inset Formula $A$
-\end_inset
-
-, 
-\begin_inset Formula $A[X]$
-\end_inset
-
- es un dominio euclídeo si y sólo si es un DIP, si y sólo si 
-\begin_inset Formula $A$
-\end_inset
-
- es un cuerpo.
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $D$
-\end_inset
-
- un dominio y 
-\begin_inset Formula $p\in D$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $p$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $D$
-\end_inset
-
- si y sólo si lo es en 
-\begin_inset Formula $D[X]$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $p$
-\end_inset
-
- es primo en 
-\begin_inset Formula $D[X]$
-\end_inset
-
-, lo es en 
-\begin_inset Formula $D$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $D$
-\end_inset
-
- es un DFU, 
-\begin_inset Formula $p$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $D$
-\end_inset
-
- si y sólo si lo es en 
-\begin_inset Formula $D[X]$
-\end_inset
-
-, si y sólo si es primo en 
-\begin_inset Formula $D$
-\end_inset
-
-, si y sólo si lo es en 
-\begin_inset Formula $D[X]$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Standard
-Sea 
-\begin_inset Formula $D$
-\end_inset
-
- un DFU, definimos 
-\begin_inset Formula $\varphi:D\setminus0\to\mathbb{N}$
-\end_inset
-
- tal que 
-\begin_inset Formula $\varphi(a)$
-\end_inset
-
- es el número de factores irreducibles en la factorización por irreducibles
- de 
-\begin_inset Formula $a$
-\end_inset
-
- en 
-\begin_inset Formula $D$
-\end_inset
-
-, contando repetidos, y para 
-\begin_inset Formula $a,b\in D\setminus\{0\}$
-\end_inset
-
-, 
-\begin_inset Formula $\varphi(ab)=\varphi(a)+\varphi(b)$
-\end_inset
-
- y 
-\begin_inset Formula $\varphi(a)=0\iff a\in D^{*}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $D$
-\end_inset
-
- es un DFU, 
-\begin_inset Formula $K$
-\end_inset
-
- es su cuerpo de fracciones y 
-\begin_inset Formula $f\in D[X]$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $D[X]$
-\end_inset
-
-, es irreducible en 
-\begin_inset Formula $K[X]$
-\end_inset
-
-.
- [...] 
-\begin_inset Formula $D$
-\end_inset
-
- es un DFU si y sólo si lo es 
-\begin_inset Formula $D[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-[...] Si 
-\begin_inset Formula $D$
-\end_inset
-
- es un DFU y 
-\begin_inset Formula $K$
-\end_inset
-
- es su cuerpo de fracciones, definimos la relación de equivalencia en 
-\begin_inset Formula $K$
-\end_inset
-
- 
-\begin_inset Formula $x\sim y:\iff\exists u\in D^{*}:y=ux$
-\end_inset
-
-, con lo que 
-\begin_inset Formula $[x]=xD^{*}$
-\end_inset
-
- y, en particular, si 
-\begin_inset Formula $x\in D$
-\end_inset
-
-, 
-\begin_inset Formula $[x]$
-\end_inset
-
- es el conjunto de los asociados de 
-\begin_inset Formula $x$
-\end_inset
-
- en 
-\begin_inset Formula $D$
-\end_inset
-
-.
- Definimos 
-\begin_inset Formula $\cdot:K\times(K/\sim)\to K/\sim$
-\end_inset
-
- como 
-\begin_inset Formula $a(bD^{*})=(ab)D^{*}$
-\end_inset
-
-.
- Esto está bien definido.
- Además, 
-\begin_inset Formula $a(b(cD^{*}))=(ab)(cD^{*})$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Definimos 
-\begin_inset Formula $c:K[X]\to K/\sim$
-\end_inset
-
- tal que, para 
-\begin_inset Formula $p\coloneqq \sum_{k\geq0}p_{k}X^{k}\in D[X]$
-\end_inset
-
-, 
-\begin_inset Formula $c(p)\coloneqq \{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$
-\end_inset
-
-, y para 
-\begin_inset Formula $p\in K[X]$
-\end_inset
-
-, si 
-\begin_inset Formula $a\in D\setminus\{0\}$
-\end_inset
-
- cumple 
-\begin_inset Formula $ap\in D[X]$
-\end_inset
-
-, 
-\begin_inset Formula $c(p)\coloneqq a^{-1}c(ap)$
-\end_inset
-
-.
- Esto está bien definido.
- Si 
-\begin_inset Formula $c(p)=aD^{*}$
-\end_inset
-
-, 
-\begin_inset Formula $a$
-\end_inset
-
- es el 
-\series bold
-contenido
-\series default
- de 
-\begin_inset Formula $p$
-\end_inset
-
- (
-\begin_inset Formula $a=c(p)$
-\end_inset
-
-).
-\end_layout
-
-\begin_layout Standard
-Para 
-\begin_inset Formula $a\in K$
-\end_inset
-
- y 
-\begin_inset Formula $p\in K[X]$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $a\in D$
-\end_inset
-
- y 
-\begin_inset Formula $p\in D[X]$
-\end_inset
-
-, 
-\begin_inset Formula $a\mid p$
-\end_inset
-
- en 
-\begin_inset Formula $D[X]$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $a\mid c(p)$
-\end_inset
-
- en 
-\begin_inset Formula $D$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $c(ap)=ac(p)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $p\in D[X]\iff c(p)\in D$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Un polinomio 
-\begin_inset Formula $p$
-\end_inset
-
- es 
-\series bold
-primitivo
-\series default
- si 
-\begin_inset Formula $c(p)=1$
-\end_inset
-
-, esto es, si 
-\begin_inset Formula $p\in D[X]$
-\end_inset
-
- y 
-\begin_inset Formula $\text{mcd}_{k}p_{k}=1$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Lema de Gauss:
-\series default
- Para 
-\begin_inset Formula $f,g\in D[X]$
-\end_inset
-
-, 
-\begin_inset Formula $c(fg)=c(f)c(g)$
-\end_inset
-
-, y en particular 
-\begin_inset Formula $fg$
-\end_inset
-
- es primitivo si y sólo si 
-\begin_inset Formula $f$
-\end_inset
-
- y 
-\begin_inset Formula $g$
-\end_inset
-
- lo son.
- [...]
-\end_layout
-
-\begin_layout Standard
-Dado 
-\begin_inset Formula $f\in D[X]\setminus D$
-\end_inset
-
- primitivo, 
-\begin_inset Formula $f$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $D[X]$
-\end_inset
-
- si y sólo si lo es en 
-\begin_inset Formula $K[X]$
-\end_inset
-
-, si y sólo si 
-\begin_inset Formula $\forall G,H\in K[X],(f=GH\implies\text{gr}(G)=0\lor\text{gr}(H)=0)$
-\end_inset
-
-, si y sólo si 
-\begin_inset Formula $\forall g,h\in D[X],(f=gh\implies\text{gr}(g)=0\lor\text{gr}(h)=0)$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Standard
-De aquí que si 
-\begin_inset Formula $D$
-\end_inset
-
- es un DFU con cuerpo de fracciones 
-\begin_inset Formula $K$
-\end_inset
-
-, los irreducibles de 
-\begin_inset Formula $D[X]$
-\end_inset
-
- son precisamente los de 
-\begin_inset Formula $D$
-\end_inset
-
- y los polinomios primitivos de 
-\begin_inset Formula $D[X]\setminus D$
-\end_inset
-
- irreducibles en 
-\begin_inset Formula $K[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-[...] Sean 
-\begin_inset Formula $K$
-\end_inset
-
- un cuerpo y 
-\begin_inset Formula $f\in K[X]$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $\text{gr}(f)=1$
-\end_inset
-
-, 
-\begin_inset Formula $f$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $K[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $\text{gr}(f)>1$
-\end_inset
-
- y 
-\begin_inset Formula $f$
-\end_inset
-
- tiene una raíz en 
-\begin_inset Formula $K$
-\end_inset
-
-, 
-\begin_inset Formula $f$
-\end_inset
-
- no es irreducible en 
-\begin_inset Formula $K[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $\text{gr}(f)\in\{2,3\}$
-\end_inset
-
-, 
-\begin_inset Formula $f$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $K[X]$
-\end_inset
-
- si y sólo si no tiene raíces en 
-\begin_inset Formula $K$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $D$
-\end_inset
-
- es un DFU con cuerpo de fracciones 
-\begin_inset Formula $K$
-\end_inset
-
-, 
-\begin_inset Formula $f\coloneqq \sum_{k}a_{k}X^{k}\in D[X]$
-\end_inset
-
- y 
-\begin_inset Formula $n\coloneqq \text{gr}(f)$
-\end_inset
-
-, todas las raíces de 
-\begin_inset Formula $f$
-\end_inset
-
- en 
-\begin_inset Formula $K$
-\end_inset
-
- son de la forma 
-\begin_inset Formula $\frac{r}{s}$
-\end_inset
-
- con 
-\begin_inset Formula $r\mid a_{0}$
-\end_inset
-
- y 
-\begin_inset Formula $s\mid a_{n}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Criterio de reducción:
-\series default
- Sean 
-\begin_inset Formula $\phi:D\to K$
-\end_inset
-
- un homomorfismo de anillos donde 
-\begin_inset Formula $D$
-\end_inset
-
- es un DFU y 
-\begin_inset Formula $K$
-\end_inset
-
- es un cuerpo, 
-\begin_inset Formula $\hat{\phi}:D[X]\to K[X]$
-\end_inset
-
- el homomorfismo inducido por 
-\begin_inset Formula $\phi$
-\end_inset
-
- y 
-\begin_inset Formula $f$
-\end_inset
-
- un polinomio primitivo de 
-\begin_inset Formula $D[X]\setminus D$
-\end_inset
-
-, si 
-\begin_inset Formula $\hat{\phi}(f)$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $K[X]$
-\end_inset
-
- y 
-\begin_inset Formula $\text{gr}(\hat{\phi}(f))=\text{gr}(f)$
-\end_inset
-
-, entonces 
-\begin_inset Formula $f$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $D[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-En particular, si 
-\begin_inset Formula $p\in\mathbb{Z}$
-\end_inset
-
- es primo, 
-\begin_inset Formula $f\coloneqq \sum_{k}a_{k}X^{k}\in\mathbb{Z}[X]$
-\end_inset
-
- es primitivo, 
-\begin_inset Formula $n\coloneqq \text{gr}(f)$
-\end_inset
-
-, 
-\begin_inset Formula $p\nmid a_{n}$
-\end_inset
-
- y 
-\begin_inset Formula $f$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $\mathbb{Z}_{p}[X]$
-\end_inset
-
-, entonces 
-\begin_inset Formula $f$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $\mathbb{Z}[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Criterio de Eisenstein:
-\series default
- Sean 
-\begin_inset Formula $D$
-\end_inset
-
- un DFU, 
-\begin_inset Formula $f\coloneqq \sum_{k}a_{k}X^{k}\in D[X]$
-\end_inset
-
- primitivo y 
-\begin_inset Formula $n\coloneqq \text{gr}f$
-\end_inset
-
-, si existe un irreducible 
-\begin_inset Formula $p\in D$
-\end_inset
-
- tal que 
-\begin_inset Formula $\forall k\in\{0,\dots,n-1\},p\mid a_{k}$
-\end_inset
-
- y 
-\begin_inset Formula $p^{2}\nmid a_{0}$
-\end_inset
-
-, entonces 
-\begin_inset Formula $f$
-\end_inset
-
- es irreducible en 
-\begin_inset Formula $D[X]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Así:
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $a\in\mathbb{Z}$
-\end_inset
-
- y existe 
-\begin_inset Formula $p\in\mathbb{Z}$
-\end_inset
-
- cuya multiplicidad en 
-\begin_inset Formula $a$
-\end_inset
-
- es 1, 
-\begin_inset Formula $X^{n}-a$
-\end_inset
-
- es irreducible.
-\end_layout
-
-\begin_layout Enumerate
-Para 
-\begin_inset Formula $n\geq3$
-\end_inset
-
-, llamamos 
-\series bold
-raíces 
-\begin_inset Formula $n$
-\end_inset
-
--ésimas de la unidad
-\series default
- o 
-\series bold
-de 1
-\series default
- a las raíces de 
-\begin_inset Formula $X^{n}-1$
-\end_inset
-
- en 
-\begin_inset Formula $\mathbb{C}$
-\end_inset
-
-, que son los 
-\begin_inset Formula $n$
-\end_inset
-
- vértices del 
-\begin_inset Formula $n$
-\end_inset
-
--ágono regular inscrito en el círculo unidad de 
-\begin_inset Formula $\mathbb{C}$
-\end_inset
-
- con un vértice en el 1.
- 
-\begin_inset Formula $X^{n}-1=(X-1)\Phi_{n}(X)$
-\end_inset
-
-, donde 
-\begin_inset Formula $\Phi_{n}(X)\coloneqq X^{n-1}+X^{n-2}+\dots+X+1$
-\end_inset
-
- es el 
-\series bold
-
-\begin_inset Formula $n$
-\end_inset
-
--ésimo polinomio ciclotómico
-\series default
- y sus raíces en 
-\begin_inset Formula $\mathbb{C}$
-\end_inset
-
- son las raíces 
-\begin_inset Formula $n$
-\end_inset
-
--ésimas de 1 distintas de 1.
- En 
-\begin_inset Formula $\mathbb{Q}$
-\end_inset
-
-, 
-\begin_inset Formula $X+1\mid\Phi_{4}(X)$
-\end_inset
-
-, pero si 
-\begin_inset Formula $n$
-\end_inset
-
- es primo, 
-\begin_inset Formula $\Phi_{n}(X)$
-\end_inset
-
- es irreducible.
-\end_layout
-
-\begin_layout Standard
-[...] Dados un anillo conmutativo 
-\begin_inset Formula $A$
-\end_inset
-
- y 
-\begin_inset Formula $n\geq2$
-\end_inset
-
-, definimos el 
-\series bold
-anillo de polinomios
-\series default
- en 
-\begin_inset Formula $n$
-\end_inset
-
- indeterminadas con coeficientes en 
-\begin_inset Formula $A$
-\end_inset
-
- como 
-\begin_inset Formula $A[X_{1},\dots,X_{n}]\coloneqq A[X_{1},\dots,X_{n-1}][X_{n}]$
-\end_inset
-
-.
- Llamamos 
-\series bold
-indeterminadas
-\series default
- a los símbolos 
-\begin_inset Formula $X_{1},\dots,X_{n}$
-\end_inset
-
- y 
-\series bold
-polinomios en 
-\begin_inset Formula $n$
-\end_inset
-
- indeterminadas
-\series default
- a los elementos de 
-\begin_inset Formula $A[X_{1},\dots,X_{n}]$
-\end_inset
-
-.
- Dados un anillo conmutativo 
-\begin_inset Formula $A$
-\end_inset
-
- y 
-\begin_inset Formula $n\in\mathbb{N}^{*}$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $A[X_{1},\dots,X_{n}]$
-\end_inset
-
- no es un cuerpo.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $A[X_{1},\dots,X_{n}]$
-\end_inset
-
- es un dominio si y sólo si lo es 
-\begin_inset Formula $A$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $A$
-\end_inset
-
- es un dominio, 
-\begin_inset Formula $A[X_{1},\dots,X_{n}]^{*}=A^{*}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $A[X_{1},\dots,X_{n}]$
-\end_inset
-
- es un DFU si y sólo si lo es 
-\begin_inset Formula $A$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $A[X_{1},\dots,X_{n}]$
-\end_inset
-
- es un DIP si y sólo si 
-\begin_inset Formula $n=1$
-\end_inset
-
- y 
-\begin_inset Formula $A$
-\end_inset
-
- es un cuerpo.
-\end_layout
-
-\begin_layout Standard
-Dados 
-\begin_inset Formula $a\in A$
-\end_inset
-
- e 
-\begin_inset Formula $i\coloneqq (i_{1},\dots,i_{n})\in\mathbb{N}^{n}$
-\end_inset
-
-, llamamos a 
-\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}\in A[X_{1},\dots,X_{n}]$
-\end_inset
-
- 
-\series bold
-monomio
-\series default
- de 
-\series bold
-tipo
-\series default
- 
-\begin_inset Formula $i$
-\end_inset
-
- y coeficiente 
-\begin_inset Formula $a$
-\end_inset
-
-.
- Todo 
-\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]$
-\end_inset
-
- se escribe de forma única como suma de monomios de distinto tipo,
-\begin_inset Formula 
-\[
-p:=\sum_{i\in\mathbb{N}^{n}}p_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}},
-\]
-
-\end_inset
-
-con 
-\begin_inset Formula $p_{i}=0$
-\end_inset
-
- para casi todo 
-\begin_inset Formula $i\in\mathbb{N}^{n}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-PUAP en 
-\begin_inset Formula $n$
-\end_inset
-
- indeterminadas:
-\series default
- Sean 
-\begin_inset Formula $A$
-\end_inset
-
- un anillo conmutativo, 
-\begin_inset Formula $n\in\mathbb{N}^{*}$
-\end_inset
-
- y 
-\begin_inset Formula $u:A\to A[X_{1},\dots,X_{n}]$
-\end_inset
-
- la inclusión:
-\end_layout
-
-\begin_layout Enumerate
-Dados un homomorfismo de anillos 
-\begin_inset Formula $f:A\to B$
-\end_inset
-
- y 
-\begin_inset Formula $b_{1},\dots,b_{n}\in B$
-\end_inset
-
-, existe un único homomorfismo de anillos 
-\begin_inset Formula $\tilde{f}:A[X_{1},\dots,X_{n}]\to B$
-\end_inset
-
- tal que 
-\begin_inset Formula $\tilde{f}\circ u=f$
-\end_inset
-
- y 
-\begin_inset Formula $\tilde{f}(X_{k})=b_{k}$
-\end_inset
-
- para 
-\begin_inset Formula $k\in\{1,\dots,n\}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Dados un anillo conmutativo 
-\begin_inset Formula $P$
-\end_inset
-
-, 
-\begin_inset Formula $T_{1},\dots,T_{n}\in P$
-\end_inset
-
- y un homomorfismo 
-\begin_inset Formula $v:A\to P$
-\end_inset
-
- tales que, dados un homomorfismo de anillos 
-\begin_inset Formula $f:A\to B$
-\end_inset
-
- y 
-\begin_inset Formula $b_{1},\dots,b_{n}\in B$
-\end_inset
-
-, existe un único homomorfismo 
-\begin_inset Formula $\tilde{f}:P\to B$
-\end_inset
-
- tal que 
-\begin_inset Formula $\tilde{f}\circ v=f$
-\end_inset
-
- y 
-\begin_inset Formula $\tilde{f}(T_{k})=b_{k}$
-\end_inset
-
- para 
-\begin_inset Formula $k\in\{1,\dots,n\}$
-\end_inset
-
-, existe un isomorfismo 
-\begin_inset Formula $\phi:A[X_{1},\dots,X_{n}]\to P$
-\end_inset
-
- tal que 
-\begin_inset Formula $\phi\circ u=v$
-\end_inset
-
- y 
-\begin_inset Formula $\phi(X_{k})=T_{k}$
-\end_inset
-
- para cada 
-\begin_inset Formula $k\in\{1,\dots,n\}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Así:
-\end_layout
-
-\begin_layout Enumerate
-Dados dos anillos conmutativos 
-\begin_inset Formula $A\subseteq B$
-\end_inset
-
- y 
-\begin_inset Formula $b_{1},\dots,b_{n}\in B$
-\end_inset
-
-, el 
-\series bold
-homomorfismo de sustitución
-\series default
- 
-\begin_inset Formula $S:A[X_{1},\dots,X_{n}]\to B$
-\end_inset
-
- viene dado por 
-\begin_inset Formula $p(b_{1},\dots,b_{n})\coloneqq S(p)\coloneqq \sum_{i\in\mathbb{N}^{n}}p_{i}b_{1}^{i_{1}}\cdots b_{n}^{i_{n}}$
-\end_inset
-
-.
- Su imagen es el subanillo de 
-\begin_inset Formula $B$
-\end_inset
-
- generado por 
-\begin_inset Formula $A\cup\{b_{1},\dots,b_{n}\}$
-\end_inset
-
-, 
-\begin_inset Formula $A[b_{1},\dots,b_{n}]$
-\end_inset
-
-, y dados dos homomorfismos de anillos 
-\begin_inset Formula $f,g:A[b_{1},\dots,b_{n}]\to C$
-\end_inset
-
-, 
-\begin_inset Formula $f=g$
-\end_inset
-
- si y sólo si 
-\begin_inset Formula $f|_{A}=g|_{A}$
-\end_inset
-
- y 
-\begin_inset Formula $f(b_{k})=g(b_{k})$
-\end_inset
-
- para todo 
-\begin_inset Formula $k$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $A$
-\end_inset
-
- un anillo y 
-\begin_inset Formula $\sigma$
-\end_inset
-
- una permutación de 
-\begin_inset Formula $\mathbb{N}_{n}$
-\end_inset
-
- con inversa 
-\begin_inset Formula $\tau\coloneqq \sigma^{-1}$
-\end_inset
-
-, tomando 
-\begin_inset Formula $B=A[X_{1},\dots,X_{n}]$
-\end_inset
-
- y 
-\begin_inset Formula $b_{k}=X_{\sigma(k)}$
-\end_inset
-
- en el punto anterior obtenemos un automorfismo 
-\begin_inset Formula $\hat{\sigma}$
-\end_inset
-
- en 
-\begin_inset Formula $A[X_{1},\dots,X_{n}]$
-\end_inset
-
- con inversa 
-\begin_inset Formula $\hat{\tau}$
-\end_inset
-
- que permuta las indeterminadas.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $A[X_{1},\dots,X_{n},Y_{1},\dots,Y_{m}]\cong A[X_{1},\dots,X_{n}][Y_{1},\dots,Y_{m}]\cong A[Y_{1},\dots,Y_{m}][X_{1},\dots,X_{n}]$
-\end_inset
-
-, por lo que en la práctica no distinguimos entre estos anillos.
-\end_layout
-
-\begin_layout Enumerate
-Todo homomorfismo de anillos conmutativos 
-\begin_inset Formula $f:A\to B$
-\end_inset
-
- induce un homomorfismo 
-\begin_inset Formula $\hat{f}:A[X_{1},\dots,X_{n}]\to B[X_{1},\dots,X_{n}]$
-\end_inset
-
- dado por 
-\begin_inset Formula $\hat{f}(p)\coloneqq \sum_{i\in\mathbb{N}^{n}}f(p_{i})X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-grado
-\series default
- de un monomio 
-\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
-\end_inset
-
- a 
-\begin_inset Formula $i_{1}+\dots+i_{n}$
-\end_inset
-
-, y grado de 
-\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]\setminus0$
-\end_inset
-
-, 
-\begin_inset Formula $\text{gr}(p)$
-\end_inset
-
-, al mayor de los grados de los monomios no nulos en la expresión por monomios
- de 
-\begin_inset Formula $p$
-\end_inset
-
-.
- Entonces 
-\begin_inset Formula $\text{gr}(p+q)\leq\max\{\text{gr}(p),\text{gr}(q)\}$
-\end_inset
-
- y 
-\begin_inset Formula $\text{gr}(pq)\leq\text{gr}(p)+\text{gr}(q)$
-\end_inset
-
-.
- 
-\end_layout
-
-\begin_layout Standard
-Un polinomio es 
-\series bold
-homogéneo
-\series default
- de grado 
-\begin_inset Formula $n$
-\end_inset
-
- si es suma de monomios de grado 
-\begin_inset Formula $n$
-\end_inset
-
-.
- Todo polinomio se escribe de modo único como suma de polinomios homogéneos
- de distintos grados, sin más que agrupar los monomios de igual grado en
- la expresión como suma de monomios.
- Así, si 
-\begin_inset Formula $D$
-\end_inset
-
- es un dominio, 
-\begin_inset Formula $\text{gr}(pq)=\text{gr}(p)+\text{gr}(q)$
-\end_inset
-
- para cualesquiera 
-\begin_inset Formula $p,q\in D[X_{1},\dots,X_{n}]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{reminder}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
 \end_body
 \end_document
diff --git a/ac/n3.lyx b/ac/n3.lyx
index 4da49b6..9f5500e 100644
--- a/ac/n3.lyx
+++ b/ac/n3.lyx
@@ -161,6 +161,55 @@ producto por escalares
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Equivalentemente, el producto currificado es un homomorfismo de anillos
+ 
+\begin_inset Formula $A\to\text{End}(M)$
+\end_inset
+
+, donde 
+\begin_inset Formula $\text{End}(M)$
+\end_inset
+
+ es el anillo de los endomorfismos del grupo abeliano 
+\begin_inset Formula $M$
+\end_inset
+
+ con la suma por componentes y la composición como producto.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Propiedades:
 \end_layout
 
@@ -304,7 +353,7 @@ anulador
 \end_inset
 
  a 
-\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M\mid Xm=0\}\leq_{A}M$
+\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M\mid Xm=0\}$
 \end_inset
 
 .
@@ -665,6 +714,10 @@ Si
 \begin_inset Formula $\text{ann}_{M}(X)\leq_{A}M$
 \end_inset
 
+, y en particular 
+\begin_inset Formula $\text{ann}_{A}(X)\trianglelefteq A$
+\end_inset
+
 .
 \end_layout
 
@@ -676,6 +729,104 @@ Si
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+8.
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $I\trianglelefteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $X\subseteq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $IX\leq_{A}M$
+\end_inset
+
+, y en particular, para 
+\begin_inset Formula $m\in M$
+\end_inset
+
+, 
+\begin_inset Formula $Im=\{bm\}_{b\in I}\leq_{A}M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $S\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $SN\leq_{A}M$
+\end_inset
+
+, y en particular, para 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, 
+\begin_inset Formula $aN=\{an\}_{n\in N}\leq_{A}M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Si 
 \begin_inset Formula $N\leq_{A}M$
 \end_inset
@@ -1109,6 +1260,103 @@ Un
 .
 \end_layout
 
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Que dos submódulos de 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ sean isomorfos no significa que lo sean los módulos cociente de 
+\begin_inset Formula $M$
+\end_inset
+
+ entre ellos, ni al revés.
+ Por ejemplo, si 
+\begin_inset Formula $_{\mathbb{Z}}M\coloneqq\mathbb{Z}_{3}\oplus\mathbb{Z}_{9}$
+\end_inset
+
+, 
+\begin_inset Formula $K\coloneqq\mathbb{Z}_{3}\oplus0$
+\end_inset
+
+, 
+\begin_inset Formula $N\coloneqq0\oplus\mathbb{Z}_{9}$
+\end_inset
+
+ y 
+\begin_inset Formula $L=((0,6))$
+\end_inset
+
+, 
+\begin_inset Formula $K\cong L$
+\end_inset
+
+ pero 
+\begin_inset Formula $\frac{M}{K}\ncong\frac{M}{L}$
+\end_inset
+
+, y 
+\begin_inset Formula $\frac{M}{K+L}\cong\frac{M}{N}$
+\end_inset
+
+ pero 
+\begin_inset Formula $K+L\ncong N$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\phi:M\to M'$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-isomorfismo, para 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $\frac{M}{N}\cong\frac{M'}{\phi(N)}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Restricción de escalares
 \end_layout
@@ -1616,6 +1864,94 @@ Finalmente, estas operaciones son inversas una de la otra, pues para
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula $W$
+\end_inset
+
+ 
+\begin_inset Formula $K$
+\end_inset
+
+-espacios vectoriales y 
+\begin_inset Formula $f:V\to V$
+\end_inset
+
+ y 
+\begin_inset Formula $g:V\to V$
+\end_inset
+
+ 
+\begin_inset Formula $K$
+\end_inset
+
+-endomorfismos, un 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-homomorfismo entre los 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulos asociados a 
+\begin_inset Formula $(V,f)$
+\end_inset
+
+ y 
+\begin_inset Formula $(W,g)$
+\end_inset
+
+ es precisamente una aplicación 
+\begin_inset Formula $K$
+\end_inset
+
+-lineal 
+\begin_inset Formula $\phi:V\to W$
+\end_inset
+
+ con 
+\begin_inset Formula $\phi\circ f=g\circ\phi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Teoremas de isomorfía
 \end_layout
@@ -1928,8 +2264,73 @@ Sea
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+clase de isomorfía
+\series default
+ es una clase de equivalencia por la relación 
+\begin_inset Quotes cld
+\end_inset
+
+ser isomorfos
+\begin_inset Quotes crd
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $I,J\trianglelefteq A$
+\end_inset
+
+, si 
+\begin_inset Formula $\frac{A}{I}\cong\frac{A}{J}$
+\end_inset
+
+ como 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos entonces 
+\begin_inset Formula $I=J$
+\end_inset
+
+, pero esto no es válido si el isomorfismo es de anillos.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
-Operaciones con submódulos
+Sistemas generadores
 \end_layout
 
 \begin_layout Standard
@@ -2291,6 +2692,265 @@ Si
 
 \end_deeper
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{M}{N}$
+\end_inset
+
+ son finitamente generados, 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+10.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $N,K\leq_{A}M$
+\end_inset
+
+, 
+\begin_inset Formula $N\cap K\eqqcolon(x_{1},\dots,x_{r})$
+\end_inset
+
+, 
+\begin_inset Formula $N+K\eqqcolon(y_{1},\dots,y_{s})$
+\end_inset
+
+ y, para 
+\begin_inset Formula $j\in\{1,\dots,s\}$
+\end_inset
+
+, 
+\begin_inset Formula $y_{j}\eqqcolon n_{j}+k_{j}$
+\end_inset
+
+ con 
+\begin_inset Formula $n_{j}\in N$
+\end_inset
+
+ y 
+\begin_inset Formula $k_{j}\in K$
+\end_inset
+
+, entonces 
+\begin_inset Formula $N=(x_{1},\dots,x_{r},n_{1},\dots,n_{s})$
+\end_inset
+
+ y 
+\begin_inset Formula $K=(x_{1},\dots,x_{r},k_{1},\dots,k_{s})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+11.
+\end_layout
+
+\end_inset
+
+Dado un entero 
+\begin_inset Formula $q\geq2$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Z}\left[\frac{1}{q}\right]=\left\{ \frac{a}{q^{n}}\right\} _{a\in\mathbb{Z},n\in\mathbb{N}}\leq_{\mathbb{Z}}\mathbb{Q}$
+\end_inset
+
+ no es finitamente generado.
+ 
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+12.
+\end_layout
+
+\end_inset
+
+Los epimorfismos conservan los conjuntos generadores.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Lema de Nakayama:
+\series default
+ Dados 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ y 
+\begin_inset Formula $J\leq A$
+\end_inset
+
+ con 
+\begin_inset Formula $J\subseteq\text{Jac}A$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado y 
+\begin_inset Formula $JM=M$
+\end_inset
+
+ entonces 
+\begin_inset Formula $M=0$
+\end_inset
+
+.
+ Esto no se cumple si 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ no es finitamente generado, pues por ejemplo 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ visto como 
+\begin_inset Formula $\mathbb{Z}_{(p)}$
+\end_inset
+
+-módulo cumple 
+\begin_inset Formula $\text{Jac}(\mathbb{Z}_{p}(\mathbb{Q}))=\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado, el único 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+ con 
+\begin_inset Formula $M=JM+N$
+\end_inset
+
+ es 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $(A,J,K)$
+\end_inset
+
+ es un anillo local, 
+\begin_inset Formula $\frac{M}{JM}$
+\end_inset
+
+ es anulado por 
+\begin_inset Formula $J$
+\end_inset
+
+ (
+\begin_inset Formula $J\subseteq\text{ann}_{A}(\frac{M}{JM})$
+\end_inset
+
+), luego es un 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial.
+ Si además 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado, 
+\begin_inset Formula $\frac{M}{JM}$
+\end_inset
+
+ es de dimensión finita, y si 
+\begin_inset Formula $_{K}\frac{M}{JM}=(\overline{m_{1}},\dots,\overline{m_{n}})$
+\end_inset
+
+ entonces 
+\begin_inset Formula $_{A}M=(m_{1},\dots,m_{n})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Sumas directas
+\end_layout
+
+\begin_layout Standard
 Sean 
 \begin_inset Formula $\{N_{i}\}_{i\in I}\subseteq{\cal L}(_{A}M)$
 \end_inset
@@ -2571,6 +3231,100 @@ La unión de un conjunto generador de
 
 \end_deeper
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $J\trianglelefteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $_{A}M=\bigoplus_{i\in I}M_{i}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Dado un 
+\begin_inset Formula $A$
+\end_inset
+
+-isomorfismo 
+\begin_inset Formula $\phi:M\to N$
+\end_inset
+
+, 
+\begin_inset Formula $N=\bigoplus_{i\in I}f(M_{i})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{ann}_{M}(J)=\bigoplus_{i\in I}\text{ann}_{M_{i}}(J)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{ann}_{A}(M)=\bigcap_{i\in I}\text{ann}_{A}(M_{i})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un DIP, 
+\begin_inset Formula $I$
+\end_inset
+
+ es finito y 
+\begin_inset Formula $\text{ann}_{A}(M_{i})=(b_{i})$
+\end_inset
+
+ para cada 
+\begin_inset Formula $i\in I$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{ann}_{A}(M)=(\text{lcm}_{i\in I}b_{i})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 \begin_inset Formula $N\leq_{A}M$
 \end_inset
 
@@ -3282,6 +4036,88 @@ TODO ejercicio Saorín 2
 
 \end_layout
 
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+8.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $e\in A$
+\end_inset
+
+ es idempotente, 
+\begin_inset Formula $eM$
+\end_inset
+
+ es sumando directo de 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $f:M\to M$
+\end_inset
+
+ es un 
+\begin_inset Formula $A$
+\end_inset
+
+-endomorfismo idempotente, 
+\begin_inset Formula $M=\ker f\oplus\text{Im}f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Módulos libres
 \end_layout
@@ -3600,7 +4436,78 @@ begin{exinfo}
 
 \end_inset
 
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+8.
+\end_layout
+
+\end_inset
+
+Los epimorfismos conservan la independencia lineal.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+9.
+\end_layout
+
+\end_inset
+
 Los isomorfismos conservan bases.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+10.
+\end_layout
+
+\end_inset
+
+Un 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+-submódulo de 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es libre si y sólo si es cíclico, si y solo si es finitamente generado.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+11.
+\end_layout
+
+\end_inset
+
+Un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo si y sólo si todo 
+\begin_inset Formula $A$
+\end_inset
+
+-módulo es libre.
+\end_layout
+
+\begin_layout Standard
 \begin_inset ERT
 status open
 
@@ -4209,6 +5116,43 @@ Sean
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $_{A}N\in{\cal L}(_{A}M)$
+\end_inset
+
+ es 
+\series bold
+finitamente cogenerado
+\series default
+ si es cocompacto.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 \begin_inset Formula $_{A}M$
 \end_inset
 
@@ -4497,6 +5441,105 @@ Como todos sus subgrupos son los de esta cadena,
 
 \end_deeper
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+5.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\frac{\mathbb{Q}}{\mathbb{Z}}=\bigoplus_{p}\mathbb{Z}_{p^{\infty}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+6.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ es noetheriano, todo 
+\begin_inset Formula $A$
+\end_inset
+
+-endomorfismo suprayectivo en 
+\begin_inset Formula $M$
+\end_inset
+
+ es inyectivo.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+7.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ es artiniano, todo 
+\begin_inset Formula $A$
+\end_inset
+
+-endomorfismo inyectivo en 
+\begin_inset Formula $M$
+\end_inset
+
+ es suprayectivo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Una 
 \series bold
 sucesión exacta corta
@@ -5032,7 +6075,7 @@ Si
 \end_inset
 
 -módulo, y en particular 
-\begin_inset Formula ${\cal L}(_{A}M)\cong{\cal L}(_{A_{1}}M_{1})\times\dots\times{\cal L}(_{A_{n}}M_{n})$
+\begin_inset Formula ${\cal L}(_{A}M)\cong\prod_{i=1}^{m}{\cal L}(_{A_{i}}M_{i})$
 \end_inset
 
 .
@@ -5306,5 +6349,398 @@ de longitud finita
 -módulo finitamente generado es de longitud finita.
 \end_layout
 
+\begin_layout Section
+Módulos y matrices
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $m,n\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal C}_{m}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal C}_{n}$
+\end_inset
+
+ las bases canónicas respectivas de los 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos libres 
+\begin_inset Formula $A^{m}$
+\end_inset
+
+ y 
+\begin_inset Formula $A^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $(f\mapsto M_{{\cal C}_{m}{\cal C}_{n}}(f)):\text{Hom}_{A}(A^{n},A^{m})\to{\cal M}_{m\times n}(A)$
+\end_inset
+
+ es un isomorfismo de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos con inversa 
+\begin_inset Formula $C\mapsto v\mapsto Cv$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula ${\cal C}_{n}\eqqcolon(e_{1},\dots,e_{n})$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal C}_{m}\eqqcolon(f_{1},\dots,f_{m})$
+\end_inset
+
+, toda 
+\begin_inset Formula $f\in\text{Hom}_{A}(A^{n},A^{m})$
+\end_inset
+
+ viene dada por los valores que le asigna a los 
+\begin_inset Formula $e_{i}$
+\end_inset
+
+, que se pueden expresar respecto a los 
+\begin_inset Formula $f_{j}$
+\end_inset
+
+ dando lugar a 
+\begin_inset Formula $M\coloneqq M_{{\cal C}_{m}{\cal C}_{n}}(f)$
+\end_inset
+
+ cuyas columnas son los 
+\begin_inset Formula $f(e_{i})$
+\end_inset
+
+, pero claramente 
+\begin_inset Formula $Me_{i}$
+\end_inset
+
+ es la 
+\begin_inset Formula $i$
+\end_inset
+
+-ésima columna de 
+\begin_inset Formula $M$
+\end_inset
+
+, y recíprocamente, si 
+\begin_inset Formula $M\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ viene dada por 
+\begin_inset Formula $f(v)\coloneqq Mv$
+\end_inset
+
+, las columnas de 
+\begin_inset Formula $M_{{\cal C}_{m}{\cal C}_{n}}(f)$
+\end_inset
+
+ son los 
+\begin_inset Formula $Me_{i}$
+\end_inset
+
+ que son las columnas de 
+\begin_inset Formula $M$
+\end_inset
+
+.
+ Que es un isomorfismo es claro tomando 
+\begin_inset Formula $(b_{ij}\coloneqq\sum_{k}a_{k}e_{k}\mapsto a_{i}f_{j})_{i,j}$
+\end_inset
+
+ como base de 
+\begin_inset Formula $\text{Hom}_{A}(A^{n},A^{m})$
+\end_inset
+
+ y viendo que conserva combinaciones lineales de los 
+\begin_inset Formula $b_{ij}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\text{GL}_{s}(K)\coloneqq\{A\in{\cal M}_{s}(K)\mid\det A\neq0\}$
+\end_inset
+
+.
+ Dada 
+\begin_inset Formula $C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+, llamamos 
+\series bold
+
+\begin_inset Formula $A$
+\end_inset
+
+-módulo asociado a 
+\begin_inset Formula $C$
+\end_inset
+
+
+\series default
+, 
+\begin_inset Formula $M(C)$
+\end_inset
+
+, a 
+\begin_inset Formula $\frac{A^{m}}{\{Cv\}_{v\in A^{n}}}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $B,C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ son 
+\series bold
+equivalentes
+\series default
+ si existen 
+\begin_inset Formula $P\in\text{GL}_{m}(A)$
+\end_inset
+
+ y 
+\begin_inset Formula $Q\in\text{GL}_{n}(A)$
+\end_inset
+
+ con 
+\begin_inset Formula $C=PBQ$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $M(B)\cong M(C)$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Se tiene 
+\begin_inset Formula $PB=CQ^{-1}$
+\end_inset
+
+, luego llamando 
+\begin_inset Formula $f_{C}:A^{n}\to A^{m}$
+\end_inset
+
+ al homomorfismo 
+\begin_inset Formula $f_{C}(v)\coloneqq Cv$
+\end_inset
+
+, 
+\begin_inset Formula $f_{P}\circ f_{B}=f_{C}\circ f_{Q^{-1}}$
+\end_inset
+
+.
+ Definiendo el homomorfismo 
+\begin_inset Formula $\psi:M(B)\to M(C)$
+\end_inset
+
+ como 
+\begin_inset Formula $\psi(\overline{a})=\overline{f_{P}(a)}$
+\end_inset
+
+, 
+\begin_inset Formula $\psi$
+\end_inset
+
+ está bien definido porque 
+\begin_inset Formula $a\in\text{Im}f_{B}\implies f_{P}(a)\in\text{Im}(f_{P}\circ f_{B})=\text{Im}(f_{C}\circ f_{Q^{-1}})=\text{Im}f_{C}$
+\end_inset
+
+, pero el homomorfismo 
+\begin_inset Formula $\phi:M(C)\to M(B)$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\phi(\overline{c})\coloneqq\overline{f_{P^{-1}}(c)}$
+\end_inset
+
+ también está bien definido porque 
+\begin_inset Formula $c\in\text{Im}f_{C}\implies f_{P^{-1}}(c)\in\text{Im}(f_{P^{-1}}\circ f_{C})=\text{Im}(f_{P^{-1}}\circ f_{C}\circ f_{Q^{-1}})=\text{Im}(f_{P})$
+\end_inset
+
+, y 
+\begin_inset Formula $\phi=\psi^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+operación
+\series default
+ o 
+\series bold
+transformación elemental por filas
+\series default
+ o 
+\series bold
+columnas
+\series default
+ en 
+\begin_inset Formula $C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ consiste en intercambiar dos filas o columnas de 
+\begin_inset Formula $C$
+\end_inset
+
+, multiplicar una por un 
+\begin_inset Formula $\alpha\in A^{*}$
+\end_inset
+
+ o sumarle a una otra multiplicada por un 
+\begin_inset Formula $\alpha\in A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{AlgL}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+matriz elemental
+\series default
+ de tamaño 
+\begin_inset Formula $n$
+\end_inset
+
+ a toda matriz obtenida al efectuar una operación elemental [...] en 
+\begin_inset Formula $I_{n}$
+\end_inset
+
+.
+ [...] Si 
+\begin_inset Formula $B$
+\end_inset
+
+ se obtiene al realizar una operación elemental por filas en 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $E$
+\end_inset
+
+ al realizar la misma en 
+\begin_inset Formula $I_{m}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $B=EA$
+\end_inset
+
+.
+ [...] Si 
+\begin_inset Formula $B$
+\end_inset
+
+ se obtiene de aplicar una operación elemental por columnas en 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $E$
+\end_inset
+
+ al aplicarla a 
+\begin_inset Formula $I_{n}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $B=AE$
+\end_inset
+
+.
+ Así, realizar una serie de estas operaciones en una matriz equivale a multiplic
+arla por uno o ambos lados por un producto de matrices elementales, el cual
+ es invertible.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Las matrices elementales son las mismas por filas que por columnas.
+ Si 
+\begin_inset Formula $B,C\in{\cal M}_{m\times n}(A)$
+\end_inset
+
+ y 
+\begin_inset Formula $C$
+\end_inset
+
+ se puede obtener aplicando a 
+\begin_inset Formula $B$
+\end_inset
+
+ una cantidad finita de transformaciones elementales por filas y por columnas,
+ entonces 
+\begin_inset Formula $B$
+\end_inset
+
+ y 
+\begin_inset Formula $C$
+\end_inset
+
+ son equivalentes, pues aplicar transformaciones por filas y columnas a
+ 
+\begin_inset Formula $B$
+\end_inset
+
+ equivale a multiplicarla a izquierda y derecha por matrices invertibles.
+\end_layout
+
 \end_body
 \end_document
diff --git a/ac/n4.lyx b/ac/n4.lyx
index 2479482..7d7e1b3 100644
--- a/ac/n4.lyx
+++ b/ac/n4.lyx
@@ -549,1176 +549,6 @@ Demostración:
 \end_layout
 
 \begin_layout Section
-Grupos abelianos
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-begin{reminder}{GyA}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-orden
-\series default
- de [un grupo] 
-\begin_inset Formula $G$
-\end_inset
-
- al cardinal del conjunto.
- [...]
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $A$
-\end_inset
-
- es un anillo, 
-\begin_inset Formula $(A,+)$
-\end_inset
-
- es su 
-\series bold
-grupo aditivo
-\series default
-, que es abeliano, y 
-\begin_inset Formula $(A^{*},\cdot)$
-\end_inset
-
- es su 
-\series bold
-grupo de unidades
-\series default
-, que es abeliano cuando el anillo es conmutativo.
- [...]
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-orden
-\series default
- de 
-\begin_inset Formula $a\in G$
-\end_inset
-
- al orden de 
-\begin_inset Formula $\langle a\rangle$
-\end_inset
-
-, 
-\begin_inset Formula $|a|\coloneqq|\langle a\rangle|$
-\end_inset
-
-, y escribimos 
-\begin_inset Formula $\langle a\rangle_{n}$
-\end_inset
-
- para referirnos a 
-\begin_inset Formula $\langle a\rangle$
-\end_inset
-
- indicando que tiene orden 
-\begin_inset Formula $n$
-\end_inset
-
-.
- El orden de 
-\begin_inset Formula $a$
-\end_inset
-
- divide al de 
-\begin_inset Formula $G$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Sea 
-\begin_inset Formula $f:\mathbb{Z}\to G$
-\end_inset
-
- el homomorfismo dado por 
-\begin_inset Formula $f(n)\coloneqq a^{n}$
-\end_inset
-
-, 
-\begin_inset Formula $\ker f=n\mathbb{Z}$
-\end_inset
-
- para algún 
-\begin_inset Formula $n\geq0$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $n=0$
-\end_inset
-
-, 
-\begin_inset Formula $f$
-\end_inset
-
- es inyectivo y 
-\begin_inset Formula $(\mathbb{Z},+)\cong\langle a\rangle$
-\end_inset
-
-, y en otro caso 
-\begin_inset Formula $\mathbb{Z}_{n}\cong\langle a\rangle$
-\end_inset
-
-, con lo que 
-\begin_inset Formula $n=|a|$
-\end_inset
-
- y 
-\begin_inset Formula $a^{n}=1\iff|a|\mid n$
-\end_inset
-
-.
- De aquí, 
-\begin_inset Formula $a^{k}=a^{l}\iff k\equiv l\bmod n$
-\end_inset
-
-, con lo que 
-\begin_inset Formula $|a|$
-\end_inset
-
- es el menor entero positivo con 
-\begin_inset Formula $a^{n}=1$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $a$
-\end_inset
-
- tiene orden finito y 
-\begin_inset Formula $n>0$
-\end_inset
-
-, 
-\begin_inset Formula 
-\[
-|a^{n}|=\frac{|a|}{\text{mcd}\{|a|,n\}}.
-\]
-
-\end_inset
-
-Si 
-\begin_inset Formula $G=\langle a\rangle$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $G$
-\end_inset
-
- tiene orden infinito, 
-\begin_inset Formula $G\cong(\mathbb{Z},+)\cong C_{\infty}$
-\end_inset
-
- y los subgrupos de 
-\begin_inset Formula $G$
-\end_inset
-
- son los 
-\begin_inset Formula $\langle a^{n}\rangle$
-\end_inset
-
- con 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $|G|=n$
-\end_inset
-
-, 
-\begin_inset Formula $G\cong(\mathbb{Z}_{n},+)\cong C_{n}$
-\end_inset
-
- y los subgrupos de 
-\begin_inset Formula $G$
-\end_inset
-
- son exactamente uno de orden 
-\begin_inset Formula $d$
-\end_inset
-
- por cada 
-\begin_inset Formula $d\mid n$
-\end_inset
-
-, 
-\begin_inset Formula $\langle a^{n/d}\rangle_{d}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Todos los subgrupos y grupos cociente de 
-\begin_inset Formula $G$
-\end_inset
-
- son cíclicos.
-\end_layout
-
-\begin_layout Standard
-Así, si 
-\begin_inset Formula $p\in\mathbb{N}$
-\end_inset
-
- es primo, todos los grupos de orden 
-\begin_inset Formula $p$
-\end_inset
-
- son isomorfos a 
-\begin_inset Formula $(\mathbb{Z}_{p},+)$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $G=\langle g_{1},\dots,g_{n}\rangle$
-\end_inset
-
- y 
-\begin_inset Formula $N\unlhd G$
-\end_inset
-
-, 
-\begin_inset Formula $G/N=\langle g_{1}N,\dots,g_{n}N\rangle$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema chino de los restos para grupos:
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $G$
-\end_inset
-
- y 
-\begin_inset Formula $H$
-\end_inset
-
- son subgrupos cíclicos de órdenes respectivos 
-\begin_inset Formula $n$
-\end_inset
-
- y 
-\begin_inset Formula $m$
-\end_inset
-
-, 
-\begin_inset Formula $G\times H$
-\end_inset
-
- es cíclico si y sólo si 
-\begin_inset Formula $n$
-\end_inset
-
- y 
-\begin_inset Formula $m$
-\end_inset
-
- son coprimos.
- [...]
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $g,h\in G$
-\end_inset
-
- tienen órdenes respectivos 
-\begin_inset Formula $n$
-\end_inset
-
- y 
-\begin_inset Formula $m$
-\end_inset
-
- coprimos y 
-\begin_inset Formula $gh=hg$
-\end_inset
-
-, entonces 
-\begin_inset Formula $\langle g,h\rangle$
-\end_inset
-
- es cíclico de orden 
-\begin_inset Formula $nm$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Standard
-Dados un grupo 
-\begin_inset Formula $G$
-\end_inset
-
- y 
-\begin_inset Formula $a\in G$
-\end_inset
-
-, llamamos 
-\series bold
-conjugado
-\series default
- de 
-\begin_inset Formula $g\in G$
-\end_inset
-
- por 
-\begin_inset Formula $a$
-\end_inset
-
- a 
-\begin_inset Formula $g^{a}\coloneqq a^{-1}ga$
-\end_inset
-
-, y conjugado de 
-\begin_inset Formula $X\subseteq G$
-\end_inset
-
- por 
-\begin_inset Formula $a$
-\end_inset
-
- a 
-\begin_inset Formula $X^{a}\coloneqq\{x^{a}\}_{x\in X}$
-\end_inset
-
-.
- Dos elementos 
-\begin_inset Formula $x,y\in G$
-\end_inset
-
- o conjuntos 
-\begin_inset Formula $x,y\subseteq G$
-\end_inset
-
- son 
-\series bold
-conjugados
-\series default
- en 
-\begin_inset Formula $G$
-\end_inset
-
- si existe 
-\begin_inset Formula $a\in G$
-\end_inset
-
- con 
-\begin_inset Formula $x^{a}=y$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $a\in G$
-\end_inset
-
-, llamamos 
-\series bold
-automorfismo interno
-\series default
- definido por 
-\begin_inset Formula $a$
-\end_inset
-
- al automorfismo 
-\begin_inset Formula $\iota_{a}:G\to G$
-\end_inset
-
- dado por 
-\begin_inset Formula $\iota_{a}(x)\coloneqq x^{a}$
-\end_inset
-
-.
- Su inverso es 
-\begin_inset Formula $\iota_{a^{-1}}$
-\end_inset
-
-.
- El conjugado por 
-\begin_inset Formula $a$
-\end_inset
-
- de un subgrupo de 
-\begin_inset Formula $G$
-\end_inset
-
- es otro subgrupo de 
-\begin_inset Formula $G$
-\end_inset
-
- del mismo orden.
- [...]
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $\forall g,a,b\in G,g^{ab}=(g^{a})^{b}$
-\end_inset
-
-, y [...] la relación de ser conjugados es de equivalencia.
- Las clases de equivalencia se llaman 
-\series bold
-clases de conjugación
-\series default
- de 
-\begin_inset Formula $G$
-\end_inset
-
-, y llamamos 
-\begin_inset Formula $a^{G}\coloneqq[a]=\{a^{g}\}_{g\in G}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Sea 
-\begin_inset Formula $X$
-\end_inset
-
- un conjunto.
- Una 
-\series bold
-acción por la izquierda
-\series default
- de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $X$
-\end_inset
-
- es una función 
-\begin_inset Formula $\cdot:G\times X\to X$
-\end_inset
-
- tal que 
-\begin_inset Formula $\forall x\in X,(\forall g,h\in G,(gh)\cdot x=g\cdot(h\cdot x)\land1\cdot x=x)$
-\end_inset
-
-, y una 
-\series bold
-acción por la derecha
-\series default
- de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $X$
-\end_inset
-
- es una función 
-\begin_inset Formula $\cdot:X\times G\to X$
-\end_inset
-
- tal que 
-\begin_inset Formula $\forall x\in X,(\forall g,h\in G,x\cdot(gh)=(x\cdot g)\cdot h\land x\cdot1=x)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Si 
-\begin_inset Formula $\cdot:G\times X\to X$
-\end_inset
-
- es una acción por la izquierda de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $X$
-\end_inset
-
- y 
-\begin_inset Formula $x\in X$
-\end_inset
-
-, llamamos 
-\series bold
-órbita
-\series default
- de 
-\begin_inset Formula $x$
-\end_inset
-
- en 
-\begin_inset Formula $G$
-\end_inset
-
- a 
-\begin_inset Formula $G\cdot x\coloneqq\{g\cdot x\}_{g\in G}$
-\end_inset
-
- y 
-\series bold
-estabilizador
-\series default
- de 
-\begin_inset Formula $x$
-\end_inset
-
- en 
-\begin_inset Formula $G$
-\end_inset
-
- a 
-\begin_inset Formula $\text{Estab}_{G}(x)\coloneqq\{g\in G\mid g\cdot x=x\}$
-\end_inset
-
-.
- Si 
-\begin_inset Formula $\cdot:X\times G\to X$
-\end_inset
-
- es una acción por la derecha de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $X$
-\end_inset
-
- y 
-\begin_inset Formula $x\in X$
-\end_inset
-
-, llamamos órbita de 
-\begin_inset Formula $x$
-\end_inset
-
- en 
-\begin_inset Formula $G$
-\end_inset
-
- a 
-\begin_inset Formula $x\cdot G\coloneqq\{x\cdot g\}_{g\in G}$
-\end_inset
-
- y estabilizador de 
-\begin_inset Formula $x$
-\end_inset
-
- en 
-\begin_inset Formula $G$
-\end_inset
-
- a 
-\begin_inset Formula $\text{Estab}_{G}(x)\coloneqq\{g\in G\mid x\cdot g=x\}$
-\end_inset
-
-.
- Las órbitas forman una partición de 
-\begin_inset Formula $G$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Llamamos 
-\series bold
-acción por traslación a la izquierda
-\series default
- a la acción por la izquierda de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $G/H$
-\end_inset
-
- dada por 
-\begin_inset Formula $g\cdot xH=gxH$
-\end_inset
-
-.
- Entonces 
-\begin_inset Formula $G\cdot xH=G/H$
-\end_inset
-
- y 
-\begin_inset Formula 
-\[
-\text{Estab}_{G}(xH)=[...]=H^{x^{-1}}.
-\]
-
-\end_inset
-
-Análogamente llamamos 
-\series bold
-acción por traslación a la derecha
-\series default
- a la acción por la derecha de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $H\backslash G$
-\end_inset
-
- dada por 
-\begin_inset Formula $Hx\cdot g=Hxg$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Cuando 
-\begin_inset Formula $H=1$
-\end_inset
-
-, la acción de traslación es de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $G$
-\end_inset
-
-, con 
-\begin_inset Formula $G\cdot x=G$
-\end_inset
-
- y 
-\begin_inset Formula $\text{Estab}_{G}(x)=1$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-La 
-\series bold
-acción por conjugación
-\series default
- de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $G$
-\end_inset
-
- es la acción por la derecha 
-\begin_inset Formula $x\cdot g\coloneqq x^{g}$
-\end_inset
-
-.
- Entonces 
-\begin_inset Formula $x\cdot G=x^{G}$
-\end_inset
-
- y 
-\begin_inset Formula $\text{Estab}_{G}(x)=C_{G}(x)$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $S$
-\end_inset
-
- es el conjunto de subgrupos de 
-\begin_inset Formula $G$
-\end_inset
-
-, la 
-\series bold
-acción por conjugación de 
-\begin_inset Formula $G$
-\end_inset
-
- en sus subgrupos
-\series default
- es la acción por la derecha de 
-\begin_inset Formula $G$
-\end_inset
-
- en 
-\begin_inset Formula $S$
-\end_inset
-
- 
-\begin_inset Formula $H\cdot g=H^{g}$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $n\in\mathbb{N}$
-\end_inset
-
- y 
-\begin_inset Formula $X$
-\end_inset
-
- es un conjunto, 
-\begin_inset Formula $\cdot:S_{n}\times X^{n}\to X^{n}$
-\end_inset
-
- dada por 
-\begin_inset Formula $\sigma\cdot(x_{1},\dots,x_{n})\coloneqq(x_{\sigma(1)},\dots,x_{\sigma(n)})$
-\end_inset
-
- es una acción por la izquierda.
-\end_layout
-
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $\cdot:G\times X\to X$
-\end_inset
-
- una acción por la izquierda, 
-\begin_inset Formula $H\leq G$
-\end_inset
-
- e 
-\begin_inset Formula $Y\subseteq X$
-\end_inset
-
-, si 
-\begin_inset Formula $\forall h\in H,y\in Y,h\cdot y\in Y$
-\end_inset
-
-, 
-\begin_inset Formula $\cdot|_{H\times Y}$
-\end_inset
-
- es una acción por la izquierda de 
-\begin_inset Formula $H$
-\end_inset
-
- en 
-\begin_inset Formula $Y$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $G$
-\end_inset
-
- un grupo actuando sobre un conjunto 
-\begin_inset Formula $X$
-\end_inset
-
-, 
-\begin_inset Formula $x\in X$
-\end_inset
-
- y 
-\begin_inset Formula $g\in G$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\text{Estab}_{G}(x)\leq G$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $[G:\text{Estab}_{G}(x)]=|G\cdot x|$
-\end_inset
-
-.
- En particular, si 
-\begin_inset Formula $G$
-\end_inset
-
- es finito, 
-\begin_inset Formula $|G\cdot x|\mid|G|$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si la acción es por la izquierda, 
-\begin_inset Formula $\text{Estab}_{G}(g\cdot x)=\text{Estab}_{G}(x)^{g^{-1}}$
-\end_inset
-
-, y si es por la derecha, 
-\begin_inset Formula $\text{Estab}_{G}(x\cdot g)=\text{Estab}_{G}(x)^{g}$
-\end_inset
-
-.
- En particular, si 
-\begin_inset Formula $x,g\in G$
-\end_inset
-
- y 
-\begin_inset Formula $H\leq G$
-\end_inset
-
-, 
-\begin_inset Formula $C_{G}(x^{g})=C_{G}(x)^{g}$
-\end_inset
-
- y 
-\begin_inset Formula $N_{G}(H^{g})=N_{G}(H)^{g}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $R$
-\end_inset
-
- es un conjunto irredundante de representantes de las órbitas, 
-\begin_inset Formula $|X|=\sum_{r\in R}|G\cdot r|=\sum_{r\in R}[G:\text{Estab}_{G}(r)]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Así, si 
-\begin_inset Formula $G$
-\end_inset
-
- es un grupo y 
-\begin_inset Formula $a\in G$
-\end_inset
-
-, 
-\begin_inset Formula $|a^{G}|=[G:C_{G}(a)]$
-\end_inset
-
-, y en particular 
-\begin_inset Formula $a^{G}$
-\end_inset
-
- es unipuntual si y sólo si 
-\begin_inset Formula $a\in Z(G)$
-\end_inset
-
-.
- 
-\series bold
-Ecuación de clases:
-\series default
- Si 
-\begin_inset Formula $G$
-\end_inset
-
- es finito y 
-\begin_inset Formula $X\subseteq G$
-\end_inset
-
- contiene exactamente un elemento de cada clase de conjugación con al menos
- dos elementos, entonces 
-\begin_inset Formula $|G|=|Z(G)|+\sum_{x\in X}[G:C_{G}(x)]$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Dado un número primo 
-\begin_inset Formula $p$
-\end_inset
-
-, un 
-\series bold
-
-\begin_inset Formula $p$
-\end_inset
-
--grupo
-\series default
- es un grupo en que todo elemento tiene orden potencia de 
-\begin_inset Formula $p$
-\end_inset
-
-, y un grupo finito es un 
-\begin_inset Formula $p$
-\end_inset
-
--grupo si y sólo si su orden es potencia de 
-\begin_inset Formula $p$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teorema de Cauchy:
-\series default
- Si 
-\begin_inset Formula $G$
-\end_inset
-
- es un grupo finito con orden múltiplo de un primo 
-\begin_inset Formula $p$
-\end_inset
-
-, 
-\begin_inset Formula $G$
-\end_inset
-
- tiene un elemento de orden 
-\begin_inset Formula $p$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Standard
-Dados un grupo finito 
-\begin_inset Formula $G$
-\end_inset
-
- y un número primo 
-\begin_inset Formula $p$
-\end_inset
-
-, 
-\begin_inset Formula $H\leq G$
-\end_inset
-
- es un 
-\series bold
-
-\begin_inset Formula $p$
-\end_inset
-
--subgrupo de Sylow
-\series default
- de 
-\begin_inset Formula $G$
-\end_inset
-
- si es un 
-\begin_inset Formula $p$
-\end_inset
-
--grupo y 
-\begin_inset Formula $[G:H]$
-\end_inset
-
- es coprimo con 
-\begin_inset Formula $p$
-\end_inset
-
-, si y sólo si es un 
-\begin_inset Formula $p$
-\end_inset
-
--grupo y 
-\begin_inset Formula $|H|$
-\end_inset
-
- es la mayor potencia de 
-\begin_inset Formula $p$
-\end_inset
-
- que divide a 
-\begin_inset Formula $|G|$
-\end_inset
-
-.
- Llamamos 
-\begin_inset Formula $s_{p}(G)$
-\end_inset
-
- al número de 
-\begin_inset Formula $p$
-\end_inset
-
--subgrupos de Sylow de 
-\begin_inset Formula $G$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Teoremas de Sylow:
-\series default
- Sean 
-\begin_inset Formula $p$
-\end_inset
-
- un número primo y 
-\begin_inset Formula $G$
-\end_inset
-
- un grupo finito de orden 
-\begin_inset Formula $n\coloneqq p^{k}m$
-\end_inset
-
- para ciertos 
-\begin_inset Formula $k,m\in\mathbb{N}$
-\end_inset
-
- con 
-\begin_inset Formula $p\nmid m$
-\end_inset
-
-.
- Entonces:
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $G$
-\end_inset
-
- tiene al menos un 
-\begin_inset Formula $p$
-\end_inset
-
--subgrupo de Sylow, que tendrá orden 
-\begin_inset Formula $p^{k}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-Si 
-\begin_inset Formula $P$
-\end_inset
-
- es un 
-\begin_inset Formula $p$
-\end_inset
-
--subgrupo de Sylow de 
-\begin_inset Formula $G$
-\end_inset
-
- y 
-\begin_inset Formula $Q$
-\end_inset
-
- es un 
-\begin_inset Formula $p$
-\end_inset
-
--subgrupo de 
-\begin_inset Formula $G$
-\end_inset
-
-, existe 
-\begin_inset Formula $g\in G$
-\end_inset
-
- tal que 
-\begin_inset Formula $Q\subseteq P^{g}$
-\end_inset
-
-.
- En particular, todos los 
-\begin_inset Formula $p$
-\end_inset
-
--subgrupos de Sylow de 
-\begin_inset Formula $G$
-\end_inset
-
- son conjugados en 
-\begin_inset Formula $G$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $s_{p}(G)\mid m$
-\end_inset
-
- y 
-\begin_inset Formula $s_{p}(G)\equiv1\bmod p$
-\end_inset
-
-.
- [...]
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{reminder}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
 Submódulos de torsión
 \end_layout
 
@@ -1776,10 +606,13 @@ submódulo de torsión
 \end_inset
 
  a 
-\begin_inset Formula $t(M)\coloneqq\{x\in M\mid x\text{ es de torsión}\}\leq_{A}M$
+\begin_inset Formula 
+\[
+t(M)\coloneqq\{x\in M\mid x\text{ es de torsión}\}\leq_{A}M.
+\]
+
 \end_inset
 
-.
  En efecto, para 
 \begin_inset Formula $a\in A$
 \end_inset
@@ -1841,10 +674,13 @@ subgrupo de
 \end_inset
 
  a 
-\begin_inset Formula $M(p)\coloneqq\{x\in M\mid x\text{ es de }p\text{-torsión}\}\leq_{A}M$
+\begin_inset Formula 
+\[
+M(p)\coloneqq\{x\in M\mid x\text{ es de }p\text{-torsión}\}\leq_{A}M.
+\]
+
 \end_inset
 
-.
  En efecto, para 
 \begin_inset Formula $a\in A$
 \end_inset
@@ -2148,7 +984,7 @@ de
 .
 \end_layout
 
-\begin_layout Enumerate
+\begin_layout Standard
 Si 
 \begin_inset Formula $G$
 \end_inset
@@ -2173,73 +1009,115 @@ Si
 .
 \end_layout
 
-\begin_layout Enumerate
-Sean 
-\begin_inset Formula $K$
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $p\in{\cal P}$
 \end_inset
 
- un cuerpo y 
-\begin_inset Formula $M_{(V,f)}$
+, 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
 \end_inset
 
- el 
-\begin_inset Formula $K[X]$
+ y 
+\begin_inset Formula $_{A}M\coloneqq\frac{A}{(p^{n})}$
 \end_inset
 
--módulo asociado a un par 
-\begin_inset Formula $(V,f)$
+, para 
+\begin_inset Formula $k\in\{0,\dots,n-1\}$
 \end_inset
 
- de un espacio vectorial y un 
-\begin_inset Formula $K$
+ es 
+\begin_inset Formula $\text{ann}_{M}(p^{k})=\frac{(p^{n-k})}{(p^{n})}$
 \end_inset
 
--endomorfismo 
-\begin_inset Formula $V\to V$
+ y para 
+\begin_inset Formula $k\geq n$
 \end_inset
 
-, 
-\begin_inset Formula $M_{(V,f)}$
+ es 
+\begin_inset Formula $\text{ann}_{M}(p^{k})=M$
 \end_inset
 
- es de torsión finitamente generado si y sólo si 
-\begin_inset Formula $_{K}V$
+, y 
+\begin_inset Formula $\text{ann}_{M}(p)$
 \end_inset
 
- es de dimensión finita, y si 
-\begin_inset Formula $p\in K[X]$
+ es un 
+\begin_inset Formula $\frac{A}{(p)}$
 \end_inset
 
- es irreducible, 
-\begin_inset Formula $M_{(V,f)}$
+-espacio vectorial de dimensión 1.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $Q$
 \end_inset
 
- es finitamente generado de 
-\begin_inset Formula $p$
+ es el cuerpo de fracciones de 
+\begin_inset Formula $A$
 \end_inset
 
--torsión si y sólo si 
-\begin_inset Formula $_{K}V$
+ y 
+\begin_inset Formula $N\leq_{A}Q$
 \end_inset
 
- es de dimensión finita y 
-\begin_inset Formula $p(f)^{m}=0\in\text{End}_{K}(V)$
+ es no nulo, 
+\begin_inset Formula $\frac{Q}{N}$
 \end_inset
 
- para cierto 
-\begin_inset Formula $m>0$
+ es un 
+\begin_inset Formula $A$
 \end_inset
 
-.
-\begin_inset Foot
+-módulo de torsión.
+\end_layout
+
+\begin_layout Standard
+Dado un 
+\begin_inset Formula $A$
+\end_inset
+
+-homomorfismo 
+\begin_inset Formula $f:M\to N$
+\end_inset
+
+, 
+\begin_inset Formula $f(t(M))\subseteq t(N)$
+\end_inset
+
+, y la inclusión puede ser estricta incluso cuando 
+\begin_inset Formula $f$
+\end_inset
+
+ es un monomorfismo o un epimorfismo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
 status open
 
 \begin_layout Plain Layout
-¿Qué será 
-\begin_inset Formula $p(f)^{m}$
-\end_inset
 
-?
+
+\backslash
+end{exinfo}
 \end_layout
 
 \end_inset
@@ -2248,7 +1126,7 @@ status open
 \end_layout
 
 \begin_layout Section
-Parte libre de torsión de un módulo finitamente generado
+Parte libre de torsión
 \end_layout
 
 \begin_layout Standard
@@ -2706,6 +1584,134 @@ Si
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula ${\cal T}$
+\end_inset
+
+ la clase de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos de torsión y 
+\begin_inset Formula ${\cal F}$
+\end_inset
+
+ la de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos libres de torsión:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $N\leq_{A}M$
+\end_inset
+
+ y tanto 
+\begin_inset Formula $N$
+\end_inset
+
+ como 
+\begin_inset Formula $\frac{N}{M}$
+\end_inset
+
+ están en una de las clases, entonces 
+\begin_inset Formula $M$
+\end_inset
+
+ también.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $N\leq_{A}M\in{\cal T}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $N,\frac{N}{M}\in{\cal T}$
+\end_inset
+
+, pero esto no se cumple para 
+\begin_inset Formula ${\cal F}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K,N\leq_{A}M$
+\end_inset
+
+ y 
+\begin_inset Formula $K+N$
+\end_inset
+
+ está en una de las clases, 
+\begin_inset Formula $K$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+ están también en la misma.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K,N\leq_{A}M$
+\end_inset
+
+ con 
+\begin_inset Formula $K,N\in{\cal T}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $K+N\in{\cal T}$
+\end_inset
+
+, pero esto no se cumple para 
+\begin_inset Formula ${\cal F}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Módulos finitamente generados de 
 \begin_inset Formula $p$
@@ -2814,7 +1820,7 @@ La existencia es por el lema de Zorn.
 status open
 
 \begin_layout Plain Layout
-TODO
+TODO ejercicio de Saorín.
 \end_layout
 
 \end_inset
@@ -2841,7 +1847,7 @@ Si
 status open
 
 \begin_layout Plain Layout
-TODO
+TODO ejercicio de Saorín.
 \end_layout
 
 \end_inset
@@ -2879,7 +1885,7 @@ Si
 status open
 
 \begin_layout Plain Layout
-TODO
+TODO ejercicio de Saorín.
 \end_layout
 
 \end_inset
@@ -3493,7 +2499,10 @@ descomposición indescomponible
 \end_inset
 
 .
- 
+\end_layout
+
+\begin_layout Standard
+
 \series bold
 Demostración:
 \series default
@@ -3899,128 +2908,228 @@ Existen enteros
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un grupo cíclico 
+\begin_inset Formula $\langle a\rangle_{n}$
+\end_inset
+
+ es indescomponible si y sólo si tiene orden potencia de primo.
+\end_layout
+
+\begin_layout Standard
+Dado un grupo 
+\begin_inset Formula $G$
+\end_inset
 
+, llamamos 
 \series bold
-Teoremas de clasificación de endomorfismos de espacios vectoriales:
+exponente
 \series default
- Sean 
-\begin_inset Formula $V$
+ o 
+\series bold
+periodo
+\series default
+ de 
+\begin_inset Formula $G$
 \end_inset
 
- un 
-\begin_inset Formula $K$
+, 
+\begin_inset Formula $\text{Exp}(G)$
 \end_inset
 
--espacio vectorial de dimensión finita y 
-\begin_inset Formula $f:V\to V$
+, al menor 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
 \end_inset
 
- un 
-\begin_inset Formula $K$
+ tal que 
+\begin_inset Formula $\forall g\in G,g^{n}=1$
+\end_inset
+
+, o a 
+\begin_inset Formula $\infty$
 \end_inset
 
--endomorfismo:
+ si este no existe.
+ [...]
 \end_layout
 
-\begin_layout Enumerate
-Existen 
-\begin_inset Formula $p_{1},\dots,p_{k}\in K[X]$
+\begin_layout Standard
+Si un grupo es finito tiene periodo finito, y si tiene periodo finito es
+ periódico.
+ Los recíprocos no se cumplen.
+ Todo 
+\begin_inset Formula $p$
 \end_inset
 
- mónicos irreducibles distintos y 
-\begin_inset Formula $n_{ij}\in\mathbb{N}^{*}$
-\end_inset
+-grupo es periódico, pero no necesariamente finito.
+ [...]
+\end_layout
 
- para 
-\begin_inset Formula $i\in\{1,\dots,k\}$
+\begin_layout Standard
+Si 
+\begin_inset Formula $A$
 \end_inset
 
- y 
-\begin_inset Formula $j\in\{1,\dots,r_{i}\}$
+ es un grupo abeliano, 
+\begin_inset Formula $B\leq A$
 \end_inset
 
-, unívocamente determinados, y vectores 
-\begin_inset Formula $v_{ij}\in V$
+, 
+\begin_inset Formula $a\in A$
 \end_inset
 
-, tales que 
-\begin_inset Formula $\bigoplus_{i=1}^{k}\bigoplus_{j=1}^{r_{i}}K\{f^{s}(v_{ij})\}_{s\geq0}$
+, 
+\begin_inset Formula $n\in\mathbb{N}$
 \end_inset
 
- es una descomposición de 
-\begin_inset Formula $V$
+ y 
+\begin_inset Formula $na=0$
 \end_inset
 
- en suma directa interna de subespacios vectoriales 
-\begin_inset Formula $f$
+, en 
+\begin_inset Formula $A/B$
 \end_inset
 
--invariantes y cada 
-\begin_inset Formula $p_{i}(f)^{n_{ij}}(v_{ij})=0\neq p_{i}(f)^{n_{ij}-1}(v_{ij})$
+ es 
+\begin_inset Formula $|a+B|\mid|a|$
 \end_inset
 
 .
+ En general estos órdenes no coinciden.
+ [...]
 \end_layout
 
-\begin_deeper
 \begin_layout Standard
-Sean 
-\begin_inset Formula $M$
+Dados dos grupos abelianos finitos 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
 \end_inset
 
- el 
-\begin_inset Formula $K[X]$
+, una descomposición por suma directa de 
+\begin_inset Formula $A$
 \end_inset
 
--módulo asociado a 
-\begin_inset Formula $(V,f)$
+ y una de 
+\begin_inset Formula $B$
 \end_inset
 
-, 
-\begin_inset Formula $W\leq V$
+ son 
+\series bold
+semejantes
+\series default
+ si existe una biyección entre los subgrupos en la descomposición de 
+\begin_inset Formula $A$
 \end_inset
 
- y 
-\begin_inset Formula $N$
+ y la de 
+\begin_inset Formula $B$
+\end_inset
+
+ que a cada subgrupo de 
+\begin_inset Formula $A$
+\end_inset
+
+ le asocia uno de 
+\begin_inset Formula $B$
+\end_inset
+
+ isomorfo.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dos grupos abelianos finitos son isomorfos si y sólo si tienen descomposiciones
+ primarias semejantes, si y sólo si tienen descomposiciones invariantes
+ semejantes, si y sólo si tienen la misma lista de divisores elementales,
+ si y sólo si tienen la misma lista de factores invariantes.
+ [...]
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
 \end_inset
 
- el 
-\begin_inset Formula $K[X]$
+
+\end_layout
+
+\begin_layout Section
+Módulos de torsión finitamente generados
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $_{A}M$
 \end_inset
 
--submódulo de 
+ es finitamente generado de torsión, llamamos 
+\series bold
+divisores irreducibles
+\series default
+ de 
 \begin_inset Formula $M$
 \end_inset
 
- asociado a 
-\begin_inset Formula $(W,f|_{W})$
+ a los 
+\begin_inset Formula $p\in{\cal P}$
 \end_inset
 
-, basta ver que 
-\begin_inset Formula $N\cong\frac{K[X]}{(p_{i}^{n_{ij}})}$
+ con 
+\begin_inset Formula $M(p)=0$
 \end_inset
 
- si y sólo si existe 
-\begin_inset Formula $v\in V$
+.
+ Si además 
+\begin_inset Formula $M\neq0$
 \end_inset
 
- tal que 
-\begin_inset Formula $W=K\{f^{s}(v)_{s\geq0}\}$
+ y sus factores invariantes son 
+\begin_inset Formula $d_{1}\mid\dots\mid d_{t}$
 \end_inset
 
- y 
-\begin_inset Formula $p_{i}(f)^{n_{ij}}(v)=0\neq p_{i}(f)^{n_{ij}-1}(v)$
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{ann}_{A}(M)=(d_{t})$
 \end_inset
 
 .
 \end_layout
 
+\begin_deeper
 \begin_layout Enumerate
 \begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
-\begin_inset Formula $\implies]$
+\begin_inset Formula $\subseteq]$
 \end_inset
 
 
@@ -4028,142 +3137,326 @@ status open
 
 \end_inset
 
-Sean 
-\begin_inset Formula $\phi:\frac{K[X]}{(p_{i}^{n_{ij}})}\to N$
+Para 
+\begin_inset Formula $a\in\text{ann}_{A}(M)$
 \end_inset
 
- el isomorfismo y 
-\begin_inset Formula $v\coloneqq\phi(\overline{1})$
+, como 
+\begin_inset Formula $\frac{A}{(d_{t})}$
+\end_inset
+
+ es isomorfo a un sumando directo de 
+\begin_inset Formula $M$
 \end_inset
 
 , 
-\begin_inset Formula $p_{i}^{n_{ij}}\overline{1}=0$
+\begin_inset Formula $a\frac{A}{(d_{t})}=0$
+\end_inset
+
+, pero 
+\begin_inset Formula $a\frac{A}{(d_{t})}=\frac{(a)+(d_{t})}{(d_{t})}=0$
 \end_inset
 
  y por tanto 
-\begin_inset Formula $0=p_{i}^{n_{ij}}\phi(\overline{1})=p_{i}^{n_{ij}}v=p_{i}(f)^{n_{ij}}(v)$
+\begin_inset Formula $(a)+(d_{t})\subseteq(d_{t})$
 \end_inset
 
- por la definición del 
-\begin_inset Formula $K[X]$
+ y 
+\begin_inset Formula $a\in(d_{t})$
 \end_inset
 
--módulo, pero 
-\begin_inset Formula $p_{i}^{n_{ij}-1}\overline{1}\neq0$
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
 \end_inset
 
- y por tanto 
-\begin_inset Formula $p_{i}(f)^{n_{ij}-1}(v_{ij})\neq0$
+
+\end_layout
+
 \end_inset
 
-.
- Finalmente, como 
-\begin_inset Formula $\frac{K[X]}{(p_{i}^{n_{ij}})}=K\{\overline{1},X\overline{1},\dots,X^{s}\overline{1},\dots\}$
+
+\begin_inset Formula $M\cong\bigoplus_{j=1}^{t}\frac{A}{(d_{j})}$
+\end_inset
+
+ y, como cada 
+\begin_inset Formula $d_{j}\mid d_{t}$
 \end_inset
 
 , 
-\begin_inset Formula $M=K\{f^{s}(v)\}_{s\geq0}$
+\begin_inset Formula $d_{t}M=0$
 \end_inset
 
- ya que 
-\begin_inset Formula $\phi(X^{s}\overline{1})=X^{s}\phi(\overline{1})=f^{s}(v)$
+, luego 
+\begin_inset Formula $(d_{t})\subseteq\text{ann}_{A}(M)$
 \end_inset
 
 .
 \end_layout
 
+\end_deeper
 \begin_layout Enumerate
-\begin_inset Argument item:1
-status open
+Un 
+\begin_inset Formula $p\in{\cal P}$
+\end_inset
 
-\begin_layout Plain Layout
-\begin_inset Formula $\impliedby]$
+ es divisor irreducible de 
+\begin_inset Formula $M$
 \end_inset
 
+ si y sólo si lo es de 
+\begin_inset Formula $d_{t}$
+\end_inset
 
+, si y sólo si existe 
+\begin_inset Formula $x\in M\setminus\{0\}$
+\end_inset
+
+ con 
+\begin_inset Formula $px=0$
+\end_inset
+
+.
 \end_layout
 
+\begin_deeper
+\begin_layout Description
+\begin_inset Formula $1\iff2]$
 \end_inset
 
-Por la hipótesis y la definición de 
-\begin_inset Formula $N$
+ Si 
+\begin_inset Formula $(p_{ij})_{1\leq i\leq k}^{1\leq j\leq r_{i}}$
+\end_inset
+
+ son los divisores elementales de 
+\begin_inset Formula $M$
 \end_inset
 
 , 
-\begin_inset Formula $N=(v)$
+\begin_inset Formula $d_{t}=p_{1}^{n_{1r_{1}}}\cdots p_{k}^{n_{kr_{k}}}$
 \end_inset
 
-, pero 
-\begin_inset Formula $v$
+, luego los divisores irreducibles son los irreducibles de la factorización
+ irreducible de 
+\begin_inset Formula $d_{t}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies3]$
 \end_inset
 
- es anulado por 
-\begin_inset Formula $p_{i}(f)^{n_{ij}}$
+ Si 
+\begin_inset Formula $M(p)\neq0$
 \end_inset
 
- y por tanto hay un epimorfismo 
-\begin_inset Formula $\psi:\frac{K[X]}{(p_{i}^{n_{ij}})}\twoheadrightarrow K[X]v=N$
+, sea 
+\begin_inset Formula $z\in M(p)\setminus\{0\}$
 \end_inset
 
  con 
-\begin_inset Formula $\ker\psi\trianglelefteq\frac{K[X]}{(p_{i}^{n_{ij}})}$
+\begin_inset Formula $\text{ann}_{A}(z)=(p^{s})$
 \end_inset
 
-, pero los únicos ideales de 
-\begin_inset Formula $\frac{K[X]}{(p_{i}^{n_{ij}})}$
+ y 
+\begin_inset Formula $s$
 \end_inset
 
- son 
-\begin_inset Formula $(\overline{p_{i}}^{k})$
+ mínimo, 
+\begin_inset Formula $s>0$
+\end_inset
+
+ ya que de lo contrario sería 
+\begin_inset Formula $(p^{s})=A$
+\end_inset
+
+ y 
+\begin_inset Formula $z=1z=0$
+\end_inset
+
+, y 
+\begin_inset Formula $x\coloneqq p^{s-1}z\in M\setminus\{0\}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $px=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ 
+\begin_inset Formula $x\in M(p)\neq0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $M$
+\end_inset
+
+ es un grupo abeliano finito, los divisores irreducibles de 
+\begin_inset Formula $M$
+\end_inset
+
+ son los 
+\begin_inset Formula $p>0$
+\end_inset
+
+ que dividen a 
+\begin_inset Formula $|M|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $_{A}M\neq0$
+\end_inset
+
+ finitamente generado de torsión, 
+\begin_inset Formula $p$
+\end_inset
+
+ un divisor irreducible de 
+\begin_inset Formula $M$
+\end_inset
+
+ y 
+\begin_inset Formula $M(p)\cong\bigoplus_{j=0}^{r}\frac{A}{(p^{n_{j}})}$
 \end_inset
 
  con 
-\begin_inset Formula $k\in\{0,\dots,n_{ij}\}$
+\begin_inset Formula $0<n_{1}\leq\dots\leq n_{r}$
 \end_inset
 
-, y como 
-\begin_inset Formula $p_{i}(f)^{n_{ij}-1}(v)\neq0$
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $0\neq\text{ann}_{M}(p_{i})\subseteq\text{ann}_{M}(p_{i}^{2})\subseteq\dots\subseteq\text{ann}_{M}(p_{i}^{s})\subseteq\dots$
 \end_inset
 
-, 
-\begin_inset Formula $\overline{p_{i}}^{n_{ij}-1}\notin\ker\psi$
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\{s\in\mathbb{N}^{*}\mid\text{ann}_{M}(p^{s})=\text{ann}_{M}(p^{s+1})\}=\{s\in\mathbb{N}^{*}\mid s\geq n_{r}\}$
 \end_inset
 
-, con lo que 
-\begin_inset Formula $\ker\psi=0$
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $s\geq n_{r}$
+\end_inset
+
+, 
+\begin_inset Formula $M(p)\subseteq\text{ann}_{M}(p^{n_{r}})\subseteq\text{ann}_{M}(p^{s})\subseteq M(p)$
 \end_inset
 
  y 
-\begin_inset Formula $\psi$
+\begin_inset Formula $\text{ann}_{M}(p^{s})=\text{ann}_{M}(p^{s+1})=M(p)$
 \end_inset
 
- es un isomorfismo.
+.
 \end_layout
 
-\end_deeper
 \begin_layout Enumerate
-Existen polinomios mónicos no constantes 
-\begin_inset Formula $d_{1}\mid\dots\mid d_{t}$
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
 \end_inset
 
- unívocamente determinados y vectores 
-\begin_inset Formula $v_{j}\in V$
+Sea 
+\begin_inset Formula $X$
+\end_inset
+
+ el conjunto de la izquierda, queremos ver que si 
+\begin_inset Formula $s\in X$
 \end_inset
 
- tales que 
-\begin_inset Formula $\bigoplus_{i=1}^{t}\text{span}\{f^{s}(v_{j})\}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+ entonces 
+\begin_inset Formula $s+1\in X$
 \end_inset
 
- es una descomposición de 
-\begin_inset Formula $V$
+, de modo que si fuera 
+\begin_inset Formula $s<n_{r}$
 \end_inset
 
- en subespacios 
-\begin_inset Formula $f$
+, por inducción sería 
+\begin_inset Formula $\text{ann}_{M}(p^{s})=\text{ann}_{M}(p^{n_{r}})=M(p)\#$
+\end_inset
+
+.
+ Sabemos que 
+\begin_inset Formula $\text{ann}_{M}(p^{s+1})\subseteq\text{ann}_{M}(p^{s+2})$
+\end_inset
+
+, y si 
+\begin_inset Formula $x\in\text{ann}_{M}(p^{s+2})$
 \end_inset
 
--invariantes y cada 
-\begin_inset Formula $d_{j}(f)(v_{j})=0$
+, 
+\begin_inset Formula $p^{s+1}(px)=0$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $px\in\text{ann}_{M}(p^{s+1})=\text{ann}_{M}(p^{s})$
+\end_inset
+
+, luego 
+\begin_inset Formula $p^{s+1}x=p^{s}(px)=0$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in\text{ann}_{M}(p^{s+1})$
+\end_inset
+
+.
+ 
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $M(p)=\text{ann}_{M}(p^{n_{r}})$
 \end_inset
 
 .
@@ -4172,723 +3465,1809 @@ Existen polinomios mónicos no constantes
 \begin_deeper
 \begin_layout Standard
 Sean 
+\begin_inset Formula $(q_{i}^{m_{ij}})_{1\leq i\leq k}^{1\leq j\leq r_{i}}$
+\end_inset
+
+ los divisores elementales de 
 \begin_inset Formula $M$
 \end_inset
 
- el 
-\begin_inset Formula $K[X]$
+ con 
+\begin_inset Formula $p=q_{1}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $r=r_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $n_{j}=m_{1j}$
+\end_inset
+
+, hay un isomorfismo 
+\begin_inset Formula $\phi:\bigoplus_{i=1}^{k}\bigoplus_{j=1}^{m_{ir_{i}}}\frac{A}{(q_{i}^{m_{ij}})}\to M$
 \end_inset
 
--módulo asociado a 
-\begin_inset Formula $(V,f)$
+, pero 
+\begin_inset Formula 
+\[
+X\coloneqq\text{ann}_{\bigoplus_{i=1}^{k}\bigoplus_{j=1}^{m_{ir_{i}}}\frac{A}{(q_{i}^{m_{ij}})}}(p^{n_{r}})=\bigoplus_{j=1}^{n_{r}}\frac{A}{(p^{n_{j}})}
+\]
+
+\end_inset
+
+ ya que, si 
+\begin_inset Formula $i\neq1$
 \end_inset
 
 , 
-\begin_inset Formula $W\leq V$
+\begin_inset Formula $\text{ann}_{\frac{A}{(q_{i}^{s})}}(p^{n_{j}})=0$
 \end_inset
 
- y 
-\begin_inset Formula $N$
+ al ser 
+\begin_inset Formula $p^{n_{j}}+(q_{i}^{s})$
 \end_inset
 
- el 
-\begin_inset Formula $K[X]$
+ una unidad de 
+\begin_inset Formula $\frac{A}{(p_{h}^{s})}$
 \end_inset
 
--submódulo de 
-\begin_inset Formula $M$
+, de modo que 
+\begin_inset Formula 
+\[
+\text{ann}_{M}(p^{n_{r}})=\phi(X)=\phi\left(\bigoplus_{j=1}^{n_{r}}\frac{A}{(p^{n_{j}})}\right)=M(p).
+\]
+
 \end_inset
 
- asociado a 
-\begin_inset Formula $(W,f|_{W})$
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $_{A}M\neq0$
 \end_inset
 
-, basta ver que 
-\begin_inset Formula $N\cong\frac{K[X]}{(d_{j})}$
+ finitamente generado de torsión, 
+\begin_inset Formula $p\in{\cal P}$
 \end_inset
 
- si y sólo si existe 
-\begin_inset Formula $v\in V$
+ un divisor irreducible de 
+\begin_inset Formula $M$
 \end_inset
 
- tal que 
-\begin_inset Formula $\{f^{s}(v)\}{}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+ y, para 
+\begin_inset Formula $h\in\mathbb{N}^{*}$
 \end_inset
 
- es base de 
-\begin_inset Formula $W$
+, 
+\begin_inset Formula $\mu_{h}$
 \end_inset
 
- como 
-\begin_inset Formula $K$
+ el número de divisores elementales de 
+\begin_inset Formula $M$
 \end_inset
 
--espacio vectorial y 
-\begin_inset Formula $d_{j}(f)(v)=0$
+ iguales a 
+\begin_inset Formula $p^{h}$
 \end_inset
 
-.
+:
 \end_layout
 
 \begin_layout Enumerate
-\begin_inset Argument item:1
-status open
+Para 
+\begin_inset Formula $h\in\mathbb{N}^{*}$
+\end_inset
 
-\begin_layout Plain Layout
-\begin_inset Formula $\implies]$
+, 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
 \end_inset
 
+ es un 
+\begin_inset Formula $\frac{A}{(p)}$
+\end_inset
 
+-espacio vectorial.
 \end_layout
 
+\begin_deeper
+\begin_layout Standard
+Como 
+\begin_inset Formula $p$
 \end_inset
 
-Sean 
-\begin_inset Formula $\phi:\frac{K[X]}{(p_{i}^{n_{ij}})}\to N$
+ es primo en un DIP, 
+\begin_inset Formula $(p)$
 \end_inset
 
- el isomorfismo y 
-\begin_inset Formula $v\coloneqq\phi(\overline{1})$
+ es maximal, luego 
+\begin_inset Formula $\frac{A}{(p)}$
+\end_inset
+
+ es un cuerpo, y el resultado se sigue de que 
+\begin_inset Formula $p$
+\end_inset
+
+ anula a 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $h\in\mathbb{N}^{*}$
+\end_inset
+
+, si 
+\begin_inset Formula $\delta_{h}\coloneqq\dim_{\frac{A}{(p)}}\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
 \end_inset
 
 , 
-\begin_inset Formula $d_{j}\overline{1}=0$
+\begin_inset Formula $\mu_{h}=\delta_{h}-\delta_{h+1}$
 \end_inset
 
- y por tanto 
-\begin_inset Formula $0=d_{j}\phi(\overline{1})=d_{j}v=d_{j}(f)(v)$
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sea 
+\begin_inset Formula $n\coloneqq\min\{s>0\mid\text{ann}_{M}(p^{s})=\text{ann}_{M}(p^{s+1})\}$
 \end_inset
 
-, y como 
-\begin_inset Formula $\frac{K[X]}{(d_{j})}=K\{\overline{1},X\overline{1},\dots,X^{\text{gr}d_{j}-1}\overline{1}\}$
+.
+ Para 
+\begin_inset Formula $h>n$
 \end_inset
 
- con 
-\begin_inset Formula $(X^{s}\overline{1})_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+, 
+\begin_inset Formula $\mu_{h}=0$
 \end_inset
 
- linealmente independiente, 
-\begin_inset Formula $N=K\{f^{s}(v)\}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+ y, como 
+\begin_inset Formula $\text{ann}_{M}(p^{s-1})=\text{ann}_{M}(p^{s})=\text{ann}_{M}(p^{s+1})$
 \end_inset
 
- con 
-\begin_inset Formula $(f^{s}(v))_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+, 
+\begin_inset Formula $\delta_{h}=\delta_{h+1}$
 \end_inset
 
- linealmente independiente.
+.
 \end_layout
 
-\begin_layout Enumerate
-\begin_inset Argument item:1
-status open
+\begin_layout Standard
+Sea ahora 
+\begin_inset Formula $h\leq n$
+\end_inset
 
-\begin_layout Plain Layout
-\begin_inset Formula $\impliedby]$
+.
+ Si 
+\begin_inset Formula $\{p=p_{1},\dots,p_{k}\}$
 \end_inset
 
+ son los divisores irreducibles (distintos) de 
+\begin_inset Formula $M$
+\end_inset
 
-\end_layout
+, entonces 
+\begin_inset Formula $\text{ann}_{M}(p^{h})=\bigoplus_{i=1}^{k}\text{ann}_{M(p_{i})}(p^{h})$
+\end_inset
 
+.
+ En efecto, si 
+\begin_inset Formula $x\in\text{ann}_{M(p_{i})}(p^{h})$
 \end_inset
 
+, 
+\begin_inset Formula $p^{h}x=0$
+\end_inset
 
-\begin_inset Formula $v$
+ en 
+\begin_inset Formula $M(p_{i})$
 \end_inset
 
- es anulado por 
-\begin_inset Formula $p_{i}(f)^{n_{ij}}$
+ y por tanto en 
+\begin_inset Formula $M$
 \end_inset
 
- y por tanto hay un epimorfismo 
-\begin_inset Formula $\psi:\frac{K[X]}{(d_{j})}\twoheadrightarrow K[X]v=K\{f^{s}(v)\}_{s\in\mathbb{N}}=K\{f^{s}(v)\}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}=N$
+, y si 
+\begin_inset Formula $x\in\text{ann}_{M}(p^{h})$
 \end_inset
 
-, pero si 
-\begin_inset Formula $p\in K[X]$
+, si 
+\begin_inset Formula $x\eqqcolon x_{1}+\dots+x_{k}$
 \end_inset
 
- con 
-\begin_inset Formula $\text{gr}p<\text{gr}d_{j}$
+ con cada 
+\begin_inset Formula $x_{i}\in M(p_{i})$
 \end_inset
 
- cumple 
-\begin_inset Formula $\psi(\overline{p})=p(f)(v)=\sum_{i}p_{i}f^{i}(v)=0$
+, entonces 
+\begin_inset Formula $0=p^{h}x=p^{h}x_{1}+\dots+p^{h}x_{k}$
 \end_inset
 
-, como los 
-\begin_inset Formula $f^{i}(v)$
+ y cada 
+\begin_inset Formula $p^{h}x_{i}=0$
 \end_inset
 
- son linealmente independiente, cada 
-\begin_inset Formula $p_{i}=0$
+, luego 
+\begin_inset Formula $x\in\bigoplus_{i=1}^{k}\text{ann}_{M(p_{i})}(p^{h})$
 \end_inset
 
- y 
-\begin_inset Formula $p=0$
+.
+ Pero para 
+\begin_inset Formula $i>1$
 \end_inset
 
-, y como cada elemento de 
-\begin_inset Formula $\frac{K[X]}{(d_{j})}$
+, si 
+\begin_inset Formula $x\in\text{ann}_{M(p_{i})}(p^{h})$
 \end_inset
 
- tiene un representante de grado menor que el de 
-\begin_inset Formula $d_{j}$
+, 
+\begin_inset Formula $p^{h}x=0$
 \end_inset
 
 , 
-\begin_inset Formula $\ker\psi=0$
+\begin_inset Formula $x\in M(p)$
 \end_inset
 
  y 
-\begin_inset Formula $\psi$
+\begin_inset Formula $x\in M(p)\cap M(p_{i})=0$
 \end_inset
 
- es un isomorfismo.
+, luego 
+\begin_inset Formula $\text{ann}_{M(p_{i})}(p^{h})=0$
+\end_inset
+
+ y queda 
+\begin_inset Formula $\text{ann}_{M}(p^{h})=\text{ann}_{M(p)}(p^{h})$
+\end_inset
+
+, con lo que podemos suponer 
+\begin_inset Formula $M=M(p)$
+\end_inset
+
+.
 \end_layout
 
-\end_deeper
 \begin_layout Standard
-\begin_inset ERT
-status open
+Para 
+\begin_inset Formula $h\in\{1,\dots,n\}$
+\end_inset
 
-\begin_layout Plain Layout
+, como
+\begin_inset Formula 
+\[
+M\cong\bigoplus_{i=1}^{n}\left(\frac{A}{(p^{i})}\right)^{\mu_{i}}\eqqcolon M'\oplus\left(\frac{A}{(p^{h})}\right)^{\mu_{h}}\oplus\dots\oplus\left(\frac{A}{(p^{n})}\right)^{\mu_{n}},
+\]
 
+\end_inset
 
-\backslash
-begin{reminder}{GyA}
-\end_layout
+se tiene
+\begin_inset Formula 
+\begin{align*}
+\text{ann}_{M}(p^{n-h}) & =M'\oplus\left(\frac{A}{(p^{h})}\right)^{\mu_{h}}\oplus\left(\frac{(p)}{(p^{h+1})}\right)^{\mu_{h+1}}\oplus\dots\oplus\left(\frac{(p^{n-h})}{(p^{n})}\right)^{\mu_{n}},\\
+\text{ann}_{M}(p^{n-h-1}) & =M'\oplus\left(\frac{(p)}{(p^{h})}\right)^{\mu_{h}}\oplus\left(\frac{(p^{2})}{(p^{h+1})}\right)^{\mu_{h+1}}\oplus\dots\oplus\left(\frac{(p^{n-h+1})}{(p^{n})}\right)^{\mu_{n}}.
+\end{align*}
 
 \end_inset
 
+El sumando directo 
+\begin_inset Formula $M''$
+\end_inset
 
-\end_layout
+ se cancela en 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{n-h})}{\text{ann}_{M}(p^{n-h-1})}$
+\end_inset
+
+ y cada 
+\begin_inset Formula 
+\[
+\frac{\left(\frac{(p^{i})}{(p^{h+i})}\right)^{\mu_{h+i}}}{\left(\frac{(p^{i+1})}{(p^{h+i})}\right)^{\mu_{h+i}}}\cong\left(\frac{(p^{i})}{(p^{i+1})}\right)^{\mu_{h+i}}\cong\left(\frac{A}{(p)}\right)^{\mu_{h+i}},
+\]
 
-\begin_layout Standard
-Un grupo cíclico 
-\begin_inset Formula $\langle a\rangle_{n}$
 \end_inset
 
- es indescomponible si y sólo si tiene orden potencia de primo.
+con lo que 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{n-h})}{\text{ann}_{M}(p^{n-h-1})}\cong\left(\frac{A}{(p)}\right)^{\mu_{h}+\mu_{h+1}+\dots+\mu_{n}}$
+\end_inset
+
+ y 
+\begin_inset Formula $\delta_{h}=\sum_{i=h}^{n}\mu_{i}$
+\end_inset
+
+, de donde se obtiene 
+\begin_inset Formula $\mu_{h}=\delta_{h}-\delta_{h+1}$
+\end_inset
+
+.
 \end_layout
 
+\end_deeper
 \begin_layout Standard
-Dado un grupo 
-\begin_inset Formula $G$
+Sean 
+\begin_inset Formula $A$
 \end_inset
 
-, llamamos 
-\series bold
-exponente
-\series default
- o 
+ un anillo arbitrario, 
+\begin_inset Formula $0=M_{0}\subseteq M_{1}\subseteq\dots\subseteq M_{n}$
+\end_inset
+
+ una cadena de 
+\begin_inset Formula $A$
+\end_inset
+
+-módulos y, para 
+\begin_inset Formula $i\in\{1,\dots,n\}$
+\end_inset
+
+, 
+\begin_inset Formula $X_{i}\subseteq M_{i}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\frac{M_{i}}{M_{i-1}}=(\overline{X_{i}})$
+\end_inset
+
+, entonces 
+\begin_inset Formula $M_{n}=(\bigcup_{i=1}^{n}X_{i})$
+\end_inset
+
+.
+ 
 \series bold
-periodo
+Demostración:
 \series default
- de 
-\begin_inset Formula $G$
+ Si 
+\begin_inset Formula $n=0$
+\end_inset
+
+ es obvio.
+ Si 
+\begin_inset Formula $n=1$
+\end_inset
+
+, la proyección canónica 
+\begin_inset Formula $M_{1}\to\frac{M_{1}}{M_{0}}$
+\end_inset
+
+ es un isomorfismo y ya estaría.
+ Si 
+\begin_inset Formula $n>1$
+\end_inset
+
+, probado esto para 
+\begin_inset Formula $n-1$
+\end_inset
+
+, para 
+\begin_inset Formula $x\in M_{n}$
 \end_inset
 
 , 
-\begin_inset Formula $\text{Exp}(G)$
+\begin_inset Formula $\overline{x}\in\frac{M_{n}}{M_{n-1}}$
 \end_inset
 
-, al menor 
-\begin_inset Formula $n\in\mathbb{N}^{*}$
+ se escribe como 
+\begin_inset Formula $\overline{x}=\sum_{y\in X_{n}}a_{y}\overline{y}$
 \end_inset
 
- tal que 
-\begin_inset Formula $\forall g\in G,g^{n}=1$
+ con los 
+\begin_inset Formula $a_{y}\in A$
 \end_inset
 
-, o a 
-\begin_inset Formula $\infty$
+ casi todos nulos, de modo que 
+\begin_inset Formula $x'\coloneqq x-\sum_{y\in X_{n}}a_{y}y\in M_{n-1}$
 \end_inset
 
- si este no existe.
- [...]
+ y, como 
+\begin_inset Formula $x'\in(\bigcup_{i=1}^{n-1}X_{i})$
+\end_inset
+
+, 
+\begin_inset Formula $x\in(\bigcup_{i=1}^{n}X_{i})$
+\end_inset
+
+.
 \end_layout
 
 \begin_layout Standard
-Si un grupo es finito tiene periodo finito, y si tiene periodo finito es
- periódico.
- Los recíprocos no se cumplen.
- Todo 
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $_{A}M$
+\end_inset
+
+ es finitamente generado de torsión, 
 \begin_inset Formula $p$
 \end_inset
 
--grupo es periódico, pero no necesariamente finito.
- [...]
+ es un divisor irreducible de 
+\begin_inset Formula $M$
+\end_inset
+
+, 
+\begin_inset Formula $n\coloneqq\min\{s\in\mathbb{N}^{*}\mid\text{ann}_{M}(p^{s})=\text{ann}_{M}(p^{s+1})\}$
+\end_inset
+
+ y 
+\begin_inset Formula $(F_{h})_{h=1}^{n}$
+\end_inset
+
+ es una familia de subconjuntos de 
+\begin_inset Formula $M(p)$
+\end_inset
+
+ tal que cada 
+\begin_inset Formula $F_{h}\subseteq\text{ann}_{M}(p^{h})$
+\end_inset
+
+ y cada 
+\begin_inset Formula $F_{h}\cup pF_{h+1}\cup\dots\cup p^{n-h}F_{n}$
+\end_inset
+
+ es una unión disjunta que induce una base de 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
+\end_inset
+
+ como 
+\begin_inset Formula $\frac{A}{(p)}$
+\end_inset
+
+-espacio vectorial:
 \end_layout
 
+\begin_layout Enumerate
+\begin_inset Formula $\forall x\in\bigcup_{i=1}^{n}F_{h},Ax\cong\frac{A}{(p^{h})}\iff x\in F_{h}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
 \begin_layout Standard
-Si 
-\begin_inset Formula $A$
+\begin_inset Formula $Ax\cong\frac{A}{(p^{h})}$
 \end_inset
 
- es un grupo abeliano, 
-\begin_inset Formula $B\leq A$
+ si y sólo si 
+\begin_inset Formula $\text{ann}_{A}(x)=p^{h}$
+\end_inset
+
+.
+ Ahora bien, si 
+\begin_inset Formula $x\in F_{h}\subseteq\text{ann}_{M}(p^{h})$
 \end_inset
 
 , 
-\begin_inset Formula $a\in A$
+\begin_inset Formula $p^{h}\in\text{ann}_{A}(x)$
+\end_inset
+
+ y 
+\begin_inset Formula $(p^{h})\subseteq\text{ann}_{A}(x)$
+\end_inset
+
+, pero si 
+\begin_inset Formula $a\in\text{ann}_{A}(x)$
+\end_inset
+
+, tomando 
+\begin_inset Formula $a\eqqcolon p^{s}b$
+\end_inset
+
+ con 
+\begin_inset Formula $s\in\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $b\nmid p$
+\end_inset
+
+, si fuera 
+\begin_inset Formula $s<h$
 \end_inset
 
 , 
-\begin_inset Formula $n\in\mathbb{N}$
+\begin_inset Formula $\overline{p^{s}x}\in p^{s}F_{h}$
+\end_inset
+
+ es elemento de una base de 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h-s})}{\text{ann}_{M}(p^{h-s-1})}$
+\end_inset
+
+, y como 
+\begin_inset Formula $ax=0$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\overline{bp^{s}x}=\overline{ax}=0$
+\end_inset
+
+, se tiene 
+\begin_inset Formula $\overline{b}=0$
 \end_inset
 
  y 
-\begin_inset Formula $na=0$
+\begin_inset Formula $b\in(p)\#$
 \end_inset
 
-, en 
-\begin_inset Formula $A/B$
+, de modo que 
+\begin_inset Formula $s\geq h$
 \end_inset
 
- es 
-\begin_inset Formula $|a+B|\mid|a|$
+, 
+\begin_inset Formula $a\in(p^{h})$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{ann}_{A}(x)\subseteq(p^{h})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $M(p)=\bigoplus_{h=1}^{n}\bigoplus_{x\in F_{h}}Ax$
 \end_inset
 
 .
- En general estos órdenes no coinciden.
- [...]
 \end_layout
 
+\begin_deeper
 \begin_layout Standard
-Dados dos grupos abelianos finitos 
-\begin_inset Formula $A$
+\begin_inset Formula $0=\text{ann}_{M}(p^{0})\subseteq\text{ann}_{M}(p^{1})\subseteq\dots\subseteq\text{ann}_{M}(p^{n})=M(p)$
+\end_inset
+
+, y si 
+\begin_inset Formula $X_{h}\coloneqq F_{h}\cup pF_{h+1}\cup\dots\cup p^{n-h}F_{n}$
+\end_inset
+
+, cada 
+\begin_inset Formula $\overline{X_{h}}$
+\end_inset
+
+ genera 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
 \end_inset
 
  y 
-\begin_inset Formula $B$
+\begin_inset Formula $X\coloneqq\bigcup_{h=1}^{n}X_{h}$
 \end_inset
 
-, una descomposición por suma directa de 
-\begin_inset Formula $A$
+ genera 
+\begin_inset Formula $M(p)$
 \end_inset
 
- y una de 
-\begin_inset Formula $B$
+, pero 
+\begin_inset Formula $X\subseteq(\bigcup_{i=1}^{n}F_{i})=\sum_{i=1}^{n}\sum_{x\in F_{i}}Ax$
 \end_inset
 
- son 
-\series bold
-semejantes
-\series default
- si existe una biyección entre los subgrupos en la descomposición de 
-\begin_inset Formula $A$
+ y por tanto 
+\begin_inset Formula $M(p)=\sum_{h=1}^{n}\sum_{x\in F_{h}}Ax$
 \end_inset
 
- y la de 
-\begin_inset Formula $B$
+.
+ Para ver que la suma es directa, si 
+\begin_inset Formula $n=1$
 \end_inset
 
- que a cada subgrupo de 
-\begin_inset Formula $A$
+, 
+\begin_inset Formula $M(p)=\text{ann}_{M}(p)$
 \end_inset
 
- le asocia uno de 
-\begin_inset Formula $B$
+ es un espacio vectorial con base 
+\begin_inset Formula $F_{1}$
 \end_inset
 
- isomorfo.
- [...]
+ ya que la proyección canónica 
+\begin_inset Formula $\text{ann}_{M}(p)\to\frac{\text{ann}_{M}(p)}{\text{ann}_{M}(1)}$
+\end_inset
+
+ es un isomorfismo.
+ Si 
+\begin_inset Formula $n>1$
+\end_inset
+
+, probado esto para 
+\begin_inset Formula $n-1$
+\end_inset
+
+, sea 
+\begin_inset Formula $\sum_{h=1}^{n}\sum_{x\in F_{h}}a_{x}x=0$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{x\in F_{n}}a_{x}x=-\sum_{h=1}^{n-1}\sum_{x\in F_{h}}a_{x}x\in(\bigcup_{h=1}^{n-1}F_{h})\subseteq\text{ann}_{M}(p^{n-1})$
+\end_inset
+
+, pero 
+\begin_inset Formula $F_{n}$
+\end_inset
+
+ induce una base del 
+\begin_inset Formula $\frac{A}{(p)}$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{n})}{\text{ann}_{M}(p^{n-1})}$
+\end_inset
+
+ y, como 
+\begin_inset Formula $\sum_{x\in F_{n}}\overline{a_{x}}\overline{x}=0\in\frac{\text{ann}_{M}(p^{n})}{\text{ann}_{M}(p^{n-1})}$
+\end_inset
+
+, cada 
+\begin_inset Formula $a_{x}\in(p)$
+\end_inset
+
+ y, llamando 
+\begin_inset Formula $a_{x}\coloneqq pa'_{x}$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{x\in F_{n}}a'_{x}(px)+\sum_{x\in\bigcup_{h=1}^{n-1}F_{h}}a_{x}x=0$
+\end_inset
+
+, pero llamando 
+\begin_inset Formula $F'_{h}\coloneqq F_{h}$
+\end_inset
+
+ para 
+\begin_inset Formula $h<n-1$
+\end_inset
+
+ y 
+\begin_inset Formula $F'_{n-1}\coloneqq F_{n-1}\cup pF_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $(F'_{h})_{h=1}^{n-1}$
+\end_inset
+
+ cumple respecto a 
+\begin_inset Formula $\text{ann}_{M}(p^{n-1})$
+\end_inset
+
+ las mismas propiedades de 
+\begin_inset Formula $(F_{h})_{h=1}^{n}$
+\end_inset
+
+ para 
+\begin_inset Formula $M(p)$
+\end_inset
+
+, y por hipótesis de inducción los submódulos 
+\begin_inset Formula $\{Apx\}_{x\in F_{n}}\cup\bigcup_{h=1}^{n-1}\{Ax\}_{x\in F_{h}}$
+\end_inset
+
+ son independientes, con lo que los 
+\begin_inset Formula $a_{x}$
+\end_inset
+
+ son todos nulos.
 \end_layout
 
-\begin_layout Standard
-Dos grupos abelianos finitos son isomorfos si y sólo si tienen descomposiciones
- primarias semejantes, si y sólo si tienen descomposiciones invariantes
- semejantes, si y sólo si tienen la misma lista de divisores elementales,
- si y sólo si tienen la misma lista de factores invariantes.
- [...]
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $|F_{h}|$
+\end_inset
+
+ es el número de divisores elementales de 
+\begin_inset Formula $M$
+\end_inset
+
+ iguales a 
+\begin_inset Formula $p^{h}$
+\end_inset
+
+.
 \end_layout
 
+\begin_deeper
 \begin_layout Standard
-\begin_inset ERT
-status open
+\begin_inset Formula $M(p)\cong\bigoplus_{h=1}^{n}\bigoplus_{x\in F_{h}}\frac{A}{(p^{h})}=\bigoplus_{h=1}^{n}\left(\frac{A}{(p^{h})}\right)^{|F_{h}|}$
+\end_inset
 
-\begin_layout Plain Layout
+.
+\end_layout
 
+\end_deeper
+\begin_layout Enumerate
+Podemos encontrar tal familia tomando una base 
+\begin_inset Formula $(\overline{x_{i}})_{i}$
+\end_inset
 
-\backslash
-end{reminder}
+ de 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
+\end_inset
+
+, haciendo 
+\begin_inset Formula $F_{n}\coloneqq\{x_{i}\}_{i}$
+\end_inset
+
+ y, para 
+\begin_inset Formula $h$
+\end_inset
+
+ de 
+\begin_inset Formula $n-1$
+\end_inset
+
+ hasta 1, completando el conjunto linealmente independiente de 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
+\end_inset
+
+ inducido por 
+\begin_inset Formula $pF_{h+1}\cup p^{2}F_{h+2}\cup\dots\cup p^{n-h}F_{n}$
+\end_inset
+
+ con vectores 
+\begin_inset Formula $(\overline{x_{i}})_{i}$
+\end_inset
+
+ para formar una base y haciendo 
+\begin_inset Formula $F_{h}\coloneqq\{x_{i}\}_{i}$
+\end_inset
+
+.
 \end_layout
 
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $h=n$
+\end_inset
+
+, la 
+\begin_inset Formula $F_{n}$
+\end_inset
+
+ definida cumple las propiedades.
+ Si 
+\begin_inset Formula $h<n$
+\end_inset
+
+ y 
+\begin_inset Formula $F_{h+1},\dots,F_{n}$
+\end_inset
+
+ cumplen las propiedades, 
+\begin_inset Formula $pF_{h+1}\cup\dots\cup p^{n-h}F_{n}$
+\end_inset
+
+ es una unión disjunta ya que, si hubiera 
+\begin_inset Formula $i,j\in\{h+1,\dots,n\}$
+\end_inset
+
+ con 
+\begin_inset Formula $i<j$
+\end_inset
+
+ y 
+\begin_inset Formula $p^{i-h}F_{i}\cap p^{j-h}F_{j}\neq\emptyset$
+\end_inset
+
+, sean 
+\begin_inset Formula $x\in F_{i}$
+\end_inset
+
+ e 
+\begin_inset Formula $y\in F_{j}$
+\end_inset
+
+ con 
+\begin_inset Formula $p^{i-h}x=p^{j-h}y$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $p^{i-h}(x-p^{j-i}y)=0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $x-p^{j-i}y\in\text{ann}_{M}(p^{j-h})\subseteq\text{ann}_{M}(p^{j-1})$
+\end_inset
+
+, pero 
+\begin_inset Formula $x$
 \end_inset
 
+ y 
+\begin_inset Formula $p^{j-i}y$
+\end_inset
+
+ son elementos de una base de 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{j})}{\text{ann}_{M}(p^{j-1})}\#$
+\end_inset
+
+.
+ Además, 
+\begin_inset Formula $\phi:\frac{\text{ann}_{M}(p^{h+1})}{\text{ann}_{M}(p^{h})}\rightarrowtail\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\phi(\overline{z})\coloneqq p\overline{z}$
+\end_inset
 
+ es un monomorfismo ya que 
+\begin_inset Formula $p\overline{z}=0\iff pz\in\text{ann}_{M}(p^{h-1})\iff z\in\text{ann}_{M}(p^{h})\iff\overline{z}=0$
+\end_inset
+
+, y como 
+\begin_inset Formula $F_{h+1}\cup\dots\cup p^{n-h-1}F_{n}$
+\end_inset
+
+ induce una base de 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h+1})}{\text{ann}_{M}(p^{h})}$
+\end_inset
+
+, 
+\begin_inset Formula $pF_{h+1}\cup\dots\cup p^{n-h}F_{n}$
+\end_inset
+
+ induce una familia linealmente independiente en 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}$
+\end_inset
+
+.
+ Completamos esta familia para formar una base y ahora la unión sigue siendo
+ disjunta por inducir una base.
 \end_layout
 
+\end_deeper
 \begin_layout Section
-Determinación de descomposiciones de módulos de torsión finitamente generados
+Descomposiciones en dominios euclídeos
 \end_layout
 
 \begin_layout Standard
-En esta sección, salvo que se indique lo contrario, 
-\begin_inset Formula $M$
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
 \end_inset
 
- es un 
-\begin_inset Formula $A$
+
+\end_layout
+
+\begin_layout Standard
+Dado un dominio 
+\begin_inset Formula $D\neq0$
 \end_inset
 
--módulo finitamente generado de torsión y 
-\begin_inset Formula $\{p_{1},\dots,p_{k}\}\coloneqq\{p\in{\cal P}\mid M(p)\neq0\}$
+, una función 
+\begin_inset Formula $\delta:D\setminus\{0\}\to\mathbb{N}$
 \end_inset
 
- son sus 
+ es 
 \series bold
-divisores irreducibles
+euclídea
 \series default
-.
- 
+ si cumple:
 \end_layout
 
-\begin_layout Standard
-Si 
-\begin_inset Formula $M\neq0$
+\begin_layout Enumerate
+\begin_inset Formula $\forall a,b\in D\setminus\{0\},(a\mid b\implies\delta(a)\leq\delta(b))$
 \end_inset
 
- tiene factores invariantes 
-\begin_inset Formula $d_{1}\mid\dots\mid d_{t}$
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D\mid(a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$
 \end_inset
 
-:
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+dominio euclídeo
+\series default
+ es uno que admite una función euclídea.
 \end_layout
 
 \begin_layout Enumerate
-\begin_inset Formula $\text{ann}_{A}(M)=(d_{t})$
+El valor absoluto es una función euclídea en 
+\begin_inset Formula $\mathbb{Z}$
 \end_inset
 
 .
 \end_layout
 
-\begin_deeper
 \begin_layout Enumerate
-\begin_inset Argument item:1
-status open
+El cuadrado del módulo complejo es una función euclídea en 
+\begin_inset Formula $\mathbb{Z}[i]$
+\end_inset
 
-\begin_layout Plain Layout
-\begin_inset Formula $\subseteq]$
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\delta$
 \end_inset
 
+ una función euclídea en 
+\begin_inset Formula $D$
+\end_inset
 
-\end_layout
+, 
+\begin_inset Formula $I$
+\end_inset
 
+ un ideal de 
+\begin_inset Formula $D$
 \end_inset
 
-Para 
-\begin_inset Formula $a\in\text{ann}_{A}(M)$
+ y 
+\begin_inset Formula $a\in I\setminus\{0\}$
 \end_inset
 
-, como 
-\begin_inset Formula $\frac{A}{(d_{t})}$
+, entonces 
+\begin_inset Formula 
+\[
+I=(a)\iff\forall x\in I\setminus\{0\},\delta(a)\leq\delta(x).
+\]
+
 \end_inset
 
- es isomorfo a un sumando directo de 
-\begin_inset Formula $M$
+
+\end_layout
+
+\begin_layout Standard
+[...] Todo dominio euclídeo es DIP.
+ Si 
+\begin_inset Formula $\delta$
 \end_inset
 
-, 
-\begin_inset Formula $a\frac{A}{(d_{t})}=0$
+ es una función euclídea en 
+\begin_inset Formula $D$
 \end_inset
 
-, pero 
-\begin_inset Formula $a\frac{A}{(d_{t})}=\frac{(a)+(d_{t})}{(d_{t})}=0$
+, un elemento 
+\begin_inset Formula $a\in D$
 \end_inset
 
- y por tanto 
-\begin_inset Formula $(a)+(d_{t})\subseteq(d_{t})$
+ es una unidad si y sólo si 
+\begin_inset Formula $\delta(a)=\delta(1)$
 \end_inset
 
- y 
-\begin_inset Formula $a\in(d_{t})$
+, si y sólo si 
+\begin_inset Formula $\forall x\in D\setminus\{0\},\delta(a)\leq\delta(x)$
 \end_inset
 
 .
 \end_layout
 
-\begin_layout Enumerate
-\begin_inset Argument item:1
+\begin_layout Standard
+\begin_inset ERT
 status open
 
 \begin_layout Plain Layout
-\begin_inset Formula $\supseteq]$
+
+
+\backslash
+end{reminder}
+\end_layout
+
 \end_inset
 
 
 \end_layout
 
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sean 
+\begin_inset Formula $A$
 \end_inset
 
+ un dominio euclídeo, 
+\begin_inset Formula $C\in{\cal M}_{m\times n}(A)$
+\end_inset
 
-\begin_inset Formula $M\cong\bigoplus_{j=1}^{t}\frac{A}{(d_{j})}$
+ y 
+\begin_inset Formula $A^{r}\oplus\bigoplus_{i=1}^{t}\frac{A}{(d_{i})}$
 \end_inset
 
- y, como cada 
-\begin_inset Formula $d_{j}\mid d_{t}$
+ la descomposición invariante externa de 
+\begin_inset Formula $M(C)$
 \end_inset
 
 , 
-\begin_inset Formula $d_{t}M=0$
+\begin_inset Formula $C$
 \end_inset
 
-, luego 
-\begin_inset Formula $(d_{t})\subseteq\text{ann}_{A}(M)$
+ es equivalente a
+\begin_inset Formula 
+\[
+\begin{pmatrix}\boxed{I_{m-r-t}}\\
+ & d_{1}\\
+ &  & \ddots\\
+ &  &  & d_{t}\\
+ &  &  &  & \phantom{0}
+\end{pmatrix}\in{\cal M}_{m\times n}(A),
+\]
+
 \end_inset
 
-.
-\end_layout
+llamada 
+\series bold
+forma normal
+\series default
+ de 
+\begin_inset Formula $C$
+\end_inset
 
-\end_deeper
-\begin_layout Enumerate
-Un 
-\begin_inset Formula $p\in{\cal P}$
+ y a la que se puede llegar desde 
+\begin_inset Formula $C$
 \end_inset
 
- es divisor irreducible de 
-\begin_inset Formula $M$
+ por transformaciones elementales.
+ 
+\series bold
+Demostración:
+\series default
+ Primero vemos que 
+\begin_inset Formula $C$
 \end_inset
 
- si y sólo si lo es de 
-\begin_inset Formula $d_{t}$
+ se puede llevar a una matriz 
+\begin_inset Formula $D$
 \end_inset
 
-, si y sólo si existe 
-\begin_inset Formula $x\in M\setminus\{0\}$
+ de la forma dada con 
+\begin_inset Formula $d_{1}\mid\dots\mid d_{t}$
+\end_inset
+
+ y luego que 
+\begin_inset Formula $M(D)$
+\end_inset
+
+ tiene la descomposición invariante indicada, y el resultado se obtiene
+ de que 
+\begin_inset Formula $M(C)\cong M(D)$
+\end_inset
+
+ y de la unicidad de la descomposición invariante.
+ Para lo primero, si 
+\begin_inset Formula $C=0$
+\end_inset
+
+, 
+\begin_inset Formula $m=0$
+\end_inset
+
+ o 
+\begin_inset Formula $n=0$
+\end_inset
+
+ no hay que hacer nada.
+ En otro caso, sean 
+\begin_inset Formula ${\cal C}\subseteq{\cal M}_{m\times n}(A)$
+\end_inset
+
+ el conjunto de matrices alcanzables desde 
+\begin_inset Formula $C$
+\end_inset
+
+ por transformaciones elementales en filas y columnas, 
+\begin_inset Formula $\delta:A\setminus\{0\}\to\mathbb{N}$
+\end_inset
+
+ una función euclídea y 
+\begin_inset Formula $X\in{\cal C}$
+\end_inset
+
+ e índices 
+\begin_inset Formula $i,j$
 \end_inset
 
  con 
-\begin_inset Formula $px=0$
+\begin_inset Formula $\delta_{0}\coloneqq\delta(X_{ij})$
+\end_inset
+
+ mínimo, por intercambio de filas 1 e 
+\begin_inset Formula $i$
+\end_inset
+
+ y de columnas 1 y 
+\begin_inset Formula $j$
+\end_inset
+
+ podemos suponer 
+\begin_inset Formula $i=j=1$
 \end_inset
 
 .
-\end_layout
+ Para 
+\begin_inset Formula $j\in\{2,\dots,n\}$
+\end_inset
 
-\begin_deeper
-\begin_layout Description
-\begin_inset Formula $1\iff2]$
+, si fuera 
+\begin_inset Formula $X_{11}\nmid X_{1j}$
 \end_inset
 
- Si 
-\begin_inset Formula $(p_{ij})_{1\leq i\leq k}^{1\leq j\leq r_{i}}$
+ sería 
+\begin_inset Formula $X_{1j}\eqqcolon qX_{11}+r$
 \end_inset
 
- son los divisores elementales de 
-\begin_inset Formula $M$
+ con 
+\begin_inset Formula $r\neq0$
 \end_inset
 
-, 
-\begin_inset Formula $d_{t}=p_{1}^{n_{1r_{1}}}\cdots p_{k}^{n_{kr_{k}}}$
+ y 
+\begin_inset Formula $\delta(r)<\delta(X_{11})$
 \end_inset
 
-, luego los divisores irreducibles son los irreducibles de la factorización
- irreducible de 
-\begin_inset Formula $d_{t}$
+, pero restando a la columna 
+\begin_inset Formula $j$
+\end_inset
+
+ la primera por 
+\begin_inset Formula $q_{1j}$
+\end_inset
+
+ quedaría una matriz 
+\begin_inset Formula $X'$
+\end_inset
+
+ con 
+\begin_inset Formula $\delta(X'_{1j})<\delta(X_{11})=\delta_{0}\#$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $X_{11}\mid X_{1j}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $j$
+\end_inset
+
+ y, análogamente, 
+\begin_inset Formula $X_{11}\mid X_{1i}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $i$
 \end_inset
 
 .
-\end_layout
+ Si ahora definimos 
+\begin_inset Formula $q_{i}$
+\end_inset
 
-\begin_layout Description
-\begin_inset Formula $1\implies3]$
+ y 
+\begin_inset Formula $s_{j}$
 \end_inset
 
- Si 
-\begin_inset Formula $M(p)\neq0$
+ de modo que cada 
+\begin_inset Formula $X_{i1}=q_{i}X_{11}$
 \end_inset
 
-, sea 
-\begin_inset Formula $z\in M(p)\setminus\{0\}$
+ y cada 
+\begin_inset Formula $X_{1j}=s_{j}X_{11}$
+\end_inset
+
+, restando a la fila 
+\begin_inset Formula $i$
+\end_inset
+
+ la primera por 
+\begin_inset Formula $q_{i}$
+\end_inset
+
+ y a la columna 
+\begin_inset Formula $j$
+\end_inset
+
+ la primera por 
+\begin_inset Formula $s_{j}$
+\end_inset
+
+ queda una matriz
+\begin_inset Formula 
+\[
+Y=\left(\begin{array}{c|c}
+X_{11} & 0\\
+\hline 0 & B
+\end{array}\right),
+\]
+
+\end_inset
+
+pero para 
+\begin_inset Formula $i,j\geq2$
+\end_inset
+
+, si fuera 
+\begin_inset Formula $X_{11}\nmid Y_{ij}$
+\end_inset
+
+, sumando a la primera fila la 
+\begin_inset Formula $i$
+\end_inset
+
+-ésima quedaría una matriz 
+\begin_inset Formula $Z$
 \end_inset
 
  con 
-\begin_inset Formula $\text{ann}_{A}(z)=(p^{s})$
+\begin_inset Formula $Z_{11}=X_{11}$
 \end_inset
 
  y 
-\begin_inset Formula $s$
+\begin_inset Formula $Z_{i1}=Y_{ij}$
 \end_inset
 
- mínimo, 
-\begin_inset Formula $s>0$
+, con lo que 
+\begin_inset Formula $Z_{i1}=qZ_{11}+r$
 \end_inset
 
- ya que de lo contrario sería 
-\begin_inset Formula $(p^{s})=A$
+ con 
+\begin_inset Formula $r\neq0$
 \end_inset
 
  y 
-\begin_inset Formula $z=1z=0$
+\begin_inset Formula $\delta(r)<\delta(Z_{11})=\delta(X_{11})=\delta_{0}$
 \end_inset
 
-, y 
-\begin_inset Formula $x\coloneqq p^{s-1}z\in M\setminus\{0\}$
+ y, restando a la 
+\begin_inset Formula $i$
 \end_inset
 
- cumple 
-\begin_inset Formula $px=0$
+-ésima fila la primera por 
+\begin_inset Formula $q$
+\end_inset
+
+, se obtendría una matriz 
+\begin_inset Formula $Z'$
+\end_inset
+
+ con 
+\begin_inset Formula $\delta(Z'_{i1})<\delta_{0}\#$
 \end_inset
 
 .
-\end_layout
+ Por tanto 
+\begin_inset Formula $X_{11}$
+\end_inset
 
-\begin_layout Description
-\begin_inset Formula $3\implies1]$
+ divide a todo elemento de 
+\begin_inset Formula $B$
 \end_inset
 
- 
-\begin_inset Formula $x\in M(p)\neq0$
+, y si 
+\begin_inset Formula $B\eqqcolon XB'$
+\end_inset
+
+, por inducción 
+\begin_inset Formula $B'$
+\end_inset
+
+ es semejante a una matriz de la forma original y por tanto 
+\begin_inset Formula $B$
+\end_inset
+
+ también lo es e 
+\begin_inset Formula $Y_{11}\mid Y_{22}\mid\dots$
+\end_inset
+
+, pero como los 
+\begin_inset Formula $Y_{ii}$
+\end_inset
+
+ nulos están al final de la 
+\begin_inset Quotes cld
+\end_inset
+
+diagonal
+\begin_inset Quotes crd
+\end_inset
+
+ y los invertibles están al principio, si hay digamos 
+\begin_inset Formula $k$
+\end_inset
+
+ invertibles, multiplicando 
+\begin_inset Formula $Y$
+\end_inset
+
+ por 
+\begin_inset Formula $\text{diag}(Y_{11}^{-1},\dots,Y_{kk}^{-1},1,\dots,1)$
+\end_inset
+
+ se obtiene la matriz 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ Para la segunda parte, sean 
+\begin_inset Formula $s\coloneqq m-r-t$
+\end_inset
+
+, 
+\begin_inset Formula $(e_{i})_{i=1}^{n}$
+\end_inset
+
+ la base canónica de 
+\begin_inset Formula $A^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $(f_{i})_{i=1}^{m}$
+\end_inset
+
+ la de 
+\begin_inset Formula $A^{m}$
+\end_inset
+
+, 
+\begin_inset Formula $f_{D}\coloneqq(v\mapsto Dv):A^{n}\to A^{m}$
+\end_inset
+
+ lleva a cada 
+\begin_inset Formula $e_{i}$
+\end_inset
+
+ a 
+\begin_inset Formula $f_{i}$
+\end_inset
+
+ para 
+\begin_inset Formula $i\in\{1,\dots,s\}$
+\end_inset
+
+, a cada 
+\begin_inset Formula $e_{s+i}$
+\end_inset
+
+ a 
+\begin_inset Formula $d_{i}f_{s+i}$
+\end_inset
+
+ para 
+\begin_inset Formula $i\in\{1,\dots,t\}$
+\end_inset
+
+ y al resto de elementos de la base canónica a 0, luego descomponiendo 
+\begin_inset Formula $A^{n}=A^{s}\oplus A^{t}\oplus A^{n-s-t}$
+\end_inset
+
+ y 
+\begin_inset Formula $A^{m}=A^{s}\oplus A^{t}\oplus A^{r}$
+\end_inset
+
+ se puede descomponer 
+\begin_inset Formula $f_{D}=1_{A^{s}}\oplus\left(\bigoplus_{i=1}^{t}(a\mapsto d_{i}a)\right)\oplus0$
+\end_inset
+
+ con 
+\begin_inset Formula $0:A^{n-s-t}\to A^{r}$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+\frac{A}{M(D)}=\frac{A}{\text{Im}f_{D}}\cong\frac{A^{s}}{A^{s}}\oplus\left(\bigoplus_{i=1}^{t}\frac{A}{(d_{i})}\right)\oplus\frac{A^{r}}{0}\cong A^{r}\oplus\bigoplus_{i=1}^{t}\frac{A}{(d_{i})}
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $d_{1}\mid\dots\mid d_{t}$
 \end_inset
 
 .
 \end_layout
 
-\end_deeper
 \begin_layout Standard
-Así, si 
-\begin_inset Formula $M$
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{samepage}
+\end_layout
+
 \end_inset
 
- es un grupo abeliano finito, los divisores irreducibles de 
-\begin_inset Formula $M$
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $A$
 \end_inset
 
- son los 
-\begin_inset Formula $p>0$
+ es un dominio euclídeo:
+\end_layout
+
+\begin_layout Enumerate
+La forma normal de 
+\begin_inset Formula $P\in\text{GL}_{k}(A)$
 \end_inset
 
- que dividen a 
-\begin_inset Formula $|M|$
+ es 
+\begin_inset Formula $I_{k}$
 \end_inset
 
 .
 \end_layout
 
+\begin_deeper
 \begin_layout Standard
-Sean 
-\begin_inset Formula $V\in_{K}\text{Vect}$
+Es de la forma 
+\begin_inset Formula $\text{diag}(1,\dots,1,d_{1},\dots,d_{t},0,\dots,0)$
 \end_inset
 
- de dimensión finita y 
-\begin_inset Formula $f\in\text{End}_{K}(V)$
+ con los 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+ no invertibles, pero es invertible y una matriz diagonal invertible debe
+ tener todos los elementos de la diagonal invertibles.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $C,D\in{\cal M}_{m\times n}(A)$
 \end_inset
 
- con polinomio característico 
-\begin_inset Formula $\varphi\in K[X]$
+ son equivalentes, es posible llegar de 
+\begin_inset Formula $C$
 \end_inset
 
-:
+ a 
+\begin_inset Formula $D$
+\end_inset
+
+ por transformaciones elementales en filas y columnas.
 \end_layout
 
-\begin_layout Enumerate
+\begin_deeper
+\begin_layout Standard
+Existen matrices invertibles 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ con 
+\begin_inset Formula $D=PCQ$
+\end_inset
+
+, pero desde 
+\begin_inset Formula $P$
+\end_inset
+
+ o 
+\begin_inset Formula $Q$
+\end_inset
+
+ se puede llegar a su forma normal, que es la identidad, por transformaciones
+ elementales, de modo que 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ son productos de matrices elementales.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
 
+\begin_layout Section
+Presentaciones de grupos abelianos finitamente generados
+\end_layout
+
+\begin_layout Standard
+Una 
 \series bold
-Teorema de Cayley-Hamilton:
+presentación
 \series default
+ de un grupo abeliano finitamente generado 
+\begin_inset Formula $M$
+\end_inset
+
+ es una expresión 
+\begin_inset Formula 
+\[
+(x_{1},\dots,x_{m}/\rho_{1},\dots,\rho_{n}),
+\]
+
+\end_inset
+
+ donde los 
+\begin_inset Formula $x_{i}$
+\end_inset
+
+ son variables o 
+\series bold
+generadores
+\series default
+ y los 
+\begin_inset Formula $\rho_{j}=\sum_{i=1}^{m}c_{ij}x_{i}$
+\end_inset
+
+ son 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+-combinaciones lineales de dichas variables o 
+\series bold
+relatores
+\series default
+, de forma que 
+\begin_inset Formula $M\cong\frac{F}{N}$
+\end_inset
+
+ siendo 
+\begin_inset Formula $F$
+\end_inset
+
+ el grupo abeliano libre con base 
+\begin_inset Formula $\{x_{1},\dots,x_{n}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+ su subgrupo generado por los 
+\begin_inset Formula $\rho_{j}$
+\end_inset
+
+, o equivalentemente, 
+\begin_inset Formula $M\cong M(C)$
+\end_inset
+
+ para 
+\begin_inset Formula $C=(c_{ij})\in{\cal M}_{m\times n}(\mathbb{Z})$
+\end_inset
+
+.
  
-\begin_inset Formula $\varphi_{f}(f)=0$
+\series bold
+Demostración:
+\series default
+ Existe un único homomorfismo 
+\begin_inset Formula $f:\mathbb{Z}^{n}\to F$
+\end_inset
+
+ que lleva cada 
+\begin_inset Formula $e_{j}$
+\end_inset
+
+ de la base canónica de 
+\begin_inset Formula $\mathbb{Z}^{n}$
+\end_inset
+
+ a 
+\begin_inset Formula $\rho_{j}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\text{Im}f=N$
+\end_inset
+
+, y un único isomorfismo 
+\begin_inset Formula $\phi:F\to\mathbb{Z}^{m}$
+\end_inset
+
+ que lleva cada 
+\begin_inset Formula $x_{i}$
+\end_inset
+
+ al elemento 
+\begin_inset Formula $\hat{e}_{i}$
+\end_inset
+
+ de la base canónica de 
+\begin_inset Formula $\mathbb{Z}^{m}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\phi\circ f$
+\end_inset
+
+ lleva cada 
+\begin_inset Formula $e_{j}$
+\end_inset
+
+ a 
+\begin_inset Formula $(c_{1j},\dots,c_{mj})$
+\end_inset
+
+ y por tanto cada 
+\begin_inset Formula $v\in\mathbb{Z}^{n}$
+\end_inset
+
+ a 
+\begin_inset Formula $Cv$
+\end_inset
+
+ y 
+\begin_inset Formula $M(C)=\frac{\mathbb{Z}^{m}}{\text{Im}(\phi\circ f)}=\frac{\phi(F)}{\phi(N)}\cong\frac{F}{N}$
 \end_inset
 
 .
 \end_layout
 
-\begin_deeper
 \begin_layout Standard
-Sean 
-\begin_inset Formula $C\in{\cal M}_{n}(K)$
+Para encontrar la estructura de un grupo abeliano finitamente generado a
+ partir de su presentación por generadores y relatores:
+\end_layout
+
+\begin_layout Enumerate
+Usar transformaciones elementales sobre la matriz 
+\begin_inset Formula $C$
 \end_inset
 
- la matriz asociada a 
-\begin_inset Formula $f$
+ asociada a la presentación hasta llegar a su forma normal 
+\begin_inset Formula $D=PCQ$
 \end_inset
 
- bajo cualquier base de 
-\begin_inset Formula $V$
+.
+\end_layout
+
+\begin_layout Enumerate
+Obtener el rango libre de torsión de 
+\begin_inset Formula $D$
 \end_inset
 
- e 
-\begin_inset Formula $I\coloneqq I_{n}$
+.
+\end_layout
+
+\begin_layout Enumerate
+Obtener los factores invariantes 
+\begin_inset Formula $d_{j}$
 \end_inset
 
-, queremos ver que 
-\begin_inset Formula $\varphi=\det(XI-C)$
+ de 
+\begin_inset Formula $D$
 \end_inset
 
- cumple 
-\begin_inset Formula $\sum_{i=0}^{n}\varphi_{i}C^{i}=0$
+ y usar el teorema chino de los restos para factorizar cada 
+\begin_inset Formula $\mathbb{Z}_{d_{j}}$
 \end_inset
 
-.
- Por la prueba de la fórmula de la matriz inversa, para toda matriz 
-\begin_inset Formula $A$
+ en producto finito de grupos abelianos de la forma 
+\begin_inset Formula $\mathbb{Z}_{p_{i}^{n_{ij}}}$
 \end_inset
 
- es 
-\begin_inset Formula $A\cdot\text{adj}(A)^{\intercal}=|A|I$
+.
+\end_layout
+
+\begin_layout Enumerate
+Una vez obtenida de aquí la descomposición primaria externa, convertirla
+ trivialmente en descomposición primaria interna de 
+\begin_inset Formula $M(D)$
 \end_inset
 
-, por lo que viendo 
-\begin_inset Formula $XI-C\in{\cal M}_{n}(K[X])$
+.
+\end_layout
+
+\begin_layout Enumerate
+Multiplicar cada sumando directo en esta descomposición por 
+\begin_inset Formula $P^{-1}$
 \end_inset
 
- es 
-\begin_inset Formula $(XI-C)\text{adj}(XI-C)^{\intercal}=\varphi I$
+, obteniendo una descomposición directa interna de 
+\begin_inset Formula $M(P^{-1}D)=M(CQ)=M(C)$
 \end_inset
 
 .
- Como las entradas de 
-\begin_inset Formula $\text{adj}(XI-C)^{\intercal}$
-\end_inset
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
 
- son polinomios de grado máximo 
-\begin_inset Formula $n-1$
 \end_inset
 
-, podemos escribir 
-\begin_inset Formula $\text{adj}(XI-C)^{t}\eqqcolon\sum_{i=0}^{n-1}B_{i}X^{i}$
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+determinante
+\series default
+ del endomorfismo 
+\begin_inset Formula $g:\mathbb{Z}^{n}\to\mathbb{Z}^{n}$
 \end_inset
 
- con cada 
-\begin_inset Formula $B_{i}\in{\cal M}_{n}(K)$
+, 
+\begin_inset Formula $\det g$
 \end_inset
 
- y entonces 
-\begin_inset Formula $(XI-C)\sum_{i=0}^{n-1}B_{i}X^{i}=\sum_{i=0}^{n}\varphi_{i}I$
+, a 
+\begin_inset Formula $\det M_{{\cal BB}}(g)$
 \end_inset
 
-.
- Viendo esta igualdad en 
-\begin_inset Formula ${\cal M}_{n}(K)[X]$
+ para cualquier base 
+\begin_inset Formula ${\cal B}$
 \end_inset
 
-, igualando coeficientes,
-\begin_inset Formula 
-\begin{align*}
-B_{n-1} & =\varphi_{n}I, & B_{n-2}-CB_{n-1} & =\varphi_{n-1}I, &  & \cdots, & B_{0}-B_{1}C & =\varphi_{1}I, & -B_{0}C & =\varphi_{0}I,
-\end{align*}
+ de 
+\begin_inset Formula $\mathbb{Z}^{n}$
+\end_inset
 
+, que no depende de la base elegida, y entonces 
+\begin_inset Formula $\frac{\mathbb{Z}^{n}}{\text{Im}g}$
 \end_inset
 
-y multiplicando la primera igualdad por 
-\begin_inset Formula $C^{n}$
+ es finito si y sólo si 
+\begin_inset Formula $\det g\neq0$
 \end_inset
 
-, la segunda por 
-\begin_inset Formula $C^{n-1}$
+, en cuyo caso su orden es el valor absoluto de 
+\begin_inset Formula $\det g$
 \end_inset
 
-, etc.,
-\begin_inset Formula 
-\begin{align*}
-C^{n}B_{n-1} & =\varphi_{n}I, & C^{n-1}B_{n-2}-C^{n}B_{n-1} & =\varphi_{n-1}I, &  & \dots,
-\end{align*}
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
 
 \end_inset
 
 
 \end_layout
 
-\end_deeper
 \end_body
 \end_document
diff --git a/ac/n5.lyx b/ac/n5.lyx
new file mode 100644
index 0000000..d9d19d9
--- /dev/null
+++ b/ac/n5.lyx
@@ -0,0 +1,4379 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo y 
+\begin_inset Formula $M$
+\end_inset
+
+ el 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulo asociado a un par 
+\begin_inset Formula $(V,f)$
+\end_inset
+
+ de un espacio vectorial y un 
+\begin_inset Formula $K$
+\end_inset
+
+-endomorfismo 
+\begin_inset Formula $V\to V$
+\end_inset
+
+, 
+\begin_inset Formula $M$
+\end_inset
+
+ es de torsión finitamente generado si y sólo si 
+\begin_inset Formula $_{K}V$
+\end_inset
+
+ es de dimensión finita, y si 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+ es irreducible, 
+\begin_inset Formula $M$
+\end_inset
+
+ es finitamente generado de 
+\begin_inset Formula $p$
+\end_inset
+
+-torsión si y sólo si 
+\begin_inset Formula $_{K}V$
+\end_inset
+
+ es de dimensión finita y 
+\begin_inset Formula $p(f)^{m}=0\in\text{End}_{K}(V)$
+\end_inset
+
+ para cierto 
+\begin_inset Formula $m>0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+En el resto de la sección, salvo que se indique lo contrario, 
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo, 
+\begin_inset Formula $V$
+\end_inset
+
+ un 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial de dimensión finita, 
+\begin_inset Formula $f:V\to V$
+\end_inset
+
+ un 
+\begin_inset Formula $K$
+\end_inset
+
+-endomorfismo y 
+\begin_inset Formula $M$
+\end_inset
+
+ el 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulo asociado a 
+\begin_inset Formula $(V,f)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teoremas de clasificación de endomorfismos de espacios vectoriales:
+\end_layout
+
+\begin_layout Enumerate
+Existen 
+\begin_inset Formula $p_{1},\dots,p_{k}\in K[X]$
+\end_inset
+
+ mónicos irreducibles distintos y 
+\begin_inset Formula $n_{ij}\in\mathbb{N}^{*}$
+\end_inset
+
+ para 
+\begin_inset Formula $i\in\{1,\dots,k\}$
+\end_inset
+
+ y 
+\begin_inset Formula $j\in\{1,\dots,r_{i}\}$
+\end_inset
+
+, unívocamente determinados, y vectores 
+\begin_inset Formula $v_{ij}\in V$
+\end_inset
+
+, tales que 
+\begin_inset Formula 
+\[
+\bigoplus_{i=1}^{k}\bigoplus_{j=1}^{r_{i}}K\{f^{s}(v_{ij})\}_{s\geq0}
+\]
+
+\end_inset
+
+es una descomposición de 
+\begin_inset Formula $V$
+\end_inset
+
+ en suma directa interna de subespacios vectoriales 
+\begin_inset Formula $f$
+\end_inset
+
+-in
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+va
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+rian
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+tes y cada 
+\begin_inset Formula $p_{i}(f)^{n_{ij}}(v_{ij})=0\neq p_{i}(f)^{n_{ij}-1}(v_{ij})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $W\leq V$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+ el 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-submódulo de 
+\begin_inset Formula $M$
+\end_inset
+
+ asociado a 
+\begin_inset Formula $(W,f|_{W})$
+\end_inset
+
+, basta ver que 
+\begin_inset Formula $N\cong\frac{K[X]}{(p_{i}^{n_{ij}})}$
+\end_inset
+
+ si y sólo si existe 
+\begin_inset Formula $v\in V$
+\end_inset
+
+ tal que 
+\begin_inset Formula $W=K\{f^{s}(v)_{s\geq0}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $p_{i}(f)^{n_{ij}}(v)=0\neq p_{i}(f)^{n_{ij}-1}(v)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $\phi:\frac{K[X]}{(p_{i}^{n_{ij}})}\to N$
+\end_inset
+
+ el isomorfismo y 
+\begin_inset Formula $v\coloneqq\phi(\overline{1})$
+\end_inset
+
+, 
+\begin_inset Formula $p_{i}^{n_{ij}}\overline{1}=0$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $0=p_{i}^{n_{ij}}\phi(\overline{1})=p_{i}^{n_{ij}}v=p_{i}(f)^{n_{ij}}(v)$
+\end_inset
+
+ por la definición del 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulo, pero 
+\begin_inset Formula $p_{i}^{n_{ij}-1}\overline{1}\neq0$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $p_{i}(f)^{n_{ij}-1}(v_{ij})\neq0$
+\end_inset
+
+.
+ Finalmente, como 
+\begin_inset Formula $\frac{K[X]}{(p_{i}^{n_{ij}})}=K\{\overline{1},X\overline{1},\dots,X^{s}\overline{1},\dots\}$
+\end_inset
+
+, 
+\begin_inset Formula $M=K\{f^{s}(v)\}_{s\geq0}$
+\end_inset
+
+ ya que 
+\begin_inset Formula $\phi(X^{s}\overline{1})=X^{s}\phi(\overline{1})=f^{s}(v)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Por la hipótesis y la definición de 
+\begin_inset Formula $N$
+\end_inset
+
+, 
+\begin_inset Formula $N=(v)$
+\end_inset
+
+, pero 
+\begin_inset Formula $v$
+\end_inset
+
+ es anulado por 
+\begin_inset Formula $p_{i}(f)^{n_{ij}}$
+\end_inset
+
+ y por tanto hay un epimorfismo 
+\begin_inset Formula $\psi:\frac{K[X]}{(p_{i}^{n_{ij}})}\twoheadrightarrow K[X]v=N$
+\end_inset
+
+ con 
+\begin_inset Formula $\ker\psi\trianglelefteq\frac{K[X]}{(p_{i}^{n_{ij}})}$
+\end_inset
+
+, pero los únicos ideales de 
+\begin_inset Formula $\frac{K[X]}{(p_{i}^{n_{ij}})}$
+\end_inset
+
+ son 
+\begin_inset Formula $(\overline{p_{i}}^{k})$
+\end_inset
+
+ con 
+\begin_inset Formula $k\in\{0,\dots,n_{ij}\}$
+\end_inset
+
+, y como 
+\begin_inset Formula $p_{i}(f)^{n_{ij}-1}(v)\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{p_{i}}^{n_{ij}-1}\notin\ker\psi$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\ker\psi=0$
+\end_inset
+
+ y 
+\begin_inset Formula $\psi$
+\end_inset
+
+ es un isomorfismo.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Existen polinomios mónicos no constantes 
+\begin_inset Formula $d_{1}\mid\dots\mid d_{t}$
+\end_inset
+
+ unívocamente determinados y vectores 
+\begin_inset Formula $v_{j}\in V$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\bigoplus_{i=1}^{t}\text{span}\{f^{s}(v_{j})\}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+\end_inset
+
+ es una descomposición de 
+\begin_inset Formula $V$
+\end_inset
+
+ en subespacios 
+\begin_inset Formula $f$
+\end_inset
+
+-invariantes y cada 
+\begin_inset Formula $d_{j}(f)(v_{j})=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $W\leq V$
+\end_inset
+
+ y 
+\begin_inset Formula $N$
+\end_inset
+
+ el 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-submódulo de 
+\begin_inset Formula $M$
+\end_inset
+
+ asociado a 
+\begin_inset Formula $(W,f|_{W})$
+\end_inset
+
+, basta ver que 
+\begin_inset Formula $N\cong\frac{K[X]}{(d_{j})}$
+\end_inset
+
+ si y sólo si existe 
+\begin_inset Formula $v\in V$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\{f^{s}(v)\}{}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $W$
+\end_inset
+
+ como 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial y 
+\begin_inset Formula $d_{j}(f)(v)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $\phi:\frac{K[X]}{(p_{i}^{n_{ij}})}\to N$
+\end_inset
+
+ el isomorfismo y 
+\begin_inset Formula $v\coloneqq\phi(\overline{1})$
+\end_inset
+
+, 
+\begin_inset Formula $d_{j}\overline{1}=0$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $0=d_{j}\phi(\overline{1})=d_{j}v=d_{j}(f)(v)$
+\end_inset
+
+, y como 
+\begin_inset Formula $\frac{K[X]}{(d_{j})}=K\{\overline{1},X\overline{1},\dots,X^{\text{gr}d_{j}-1}\overline{1}\}$
+\end_inset
+
+ con 
+\begin_inset Formula $(X^{s}\overline{1})_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+\end_inset
+
+ linealmente independiente, 
+\begin_inset Formula $N=K\{f^{s}(v)\}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+\end_inset
+
+ con 
+\begin_inset Formula $(f^{s}(v))_{s\in\mathbb{N}_{\text{gr}(d_{j})}}$
+\end_inset
+
+ linealmente independiente.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $v$
+\end_inset
+
+ es anulado por 
+\begin_inset Formula $p_{i}(f)^{n_{ij}}$
+\end_inset
+
+ y por tanto hay un epimorfismo 
+\begin_inset Formula $\psi:\frac{K[X]}{(d_{j})}\twoheadrightarrow K[X]v=K\{f^{s}(v)\}_{s\in\mathbb{N}}=K\{f^{s}(v)\}_{s\in\mathbb{N}_{\text{gr}(d_{j})}}=N$
+\end_inset
+
+, pero si 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{gr}p<\text{gr}d_{j}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $\psi(\overline{p})=p(f)(v)=\sum_{i}p_{i}f^{i}(v)=0$
+\end_inset
+
+, como los 
+\begin_inset Formula $f^{i}(v)$
+\end_inset
+
+ son linealmente independiente, cada 
+\begin_inset Formula $p_{i}=0$
+\end_inset
+
+ y 
+\begin_inset Formula $p=0$
+\end_inset
+
+, y como cada elemento de 
+\begin_inset Formula $\frac{K[X]}{(d_{j})}$
+\end_inset
+
+ tiene un representante de grado menor que el de 
+\begin_inset Formula $d_{j}$
+\end_inset
+
+, 
+\begin_inset Formula $\ker\psi=0$
+\end_inset
+
+ y 
+\begin_inset Formula $\psi$
+\end_inset
+
+ es un isomorfismo.
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Polinomio mínimo
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\varphi\in K[X]$
+\end_inset
+
+ el polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema de Cayley-Hamilton:
+\series default
+ 
+\begin_inset Formula $\varphi(f)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $C\in{\cal M}_{n}(K)$
+\end_inset
+
+ la matriz asociada a 
+\begin_inset Formula $f$
+\end_inset
+
+ bajo cualquier base de 
+\begin_inset Formula $V$
+\end_inset
+
+ e 
+\begin_inset Formula $I\coloneqq I_{n}$
+\end_inset
+
+, queremos ver que 
+\begin_inset Formula $\varphi=\det(XI-C)$
+\end_inset
+
+ cumple 
+\begin_inset Formula $\sum_{i=0}^{n}\varphi_{i}C^{i}=0$
+\end_inset
+
+.
+ Por la prueba de la fórmula de la matriz inversa, para toda matriz 
+\begin_inset Formula $A$
+\end_inset
+
+ es 
+\begin_inset Formula $A\cdot\text{adj}(A)^{\intercal}=|A|I$
+\end_inset
+
+, por lo que viendo 
+\begin_inset Formula $XI-C\in{\cal M}_{n}(K[X])$
+\end_inset
+
+ es 
+\begin_inset Formula $(XI-C)\text{adj}(XI-C)^{\intercal}=\varphi I$
+\end_inset
+
+.
+ Como las entradas de 
+\begin_inset Formula $\text{adj}(XI-C)^{\intercal}$
+\end_inset
+
+ son polinomios de grado máximo 
+\begin_inset Formula $n-1$
+\end_inset
+
+, podemos escribir 
+\begin_inset Formula $\text{adj}(XI-C)^{t}\eqqcolon\sum_{i=0}^{n-1}B_{i}X^{i}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $B_{i}\in{\cal M}_{n}(K)$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $(XI-C)\sum_{i=0}^{n-1}B_{i}X^{i}=\sum_{i=0}^{n}\varphi_{i}I$
+\end_inset
+
+.
+ Viendo esta igualdad en 
+\begin_inset Formula ${\cal M}_{n}(K)[X]$
+\end_inset
+
+, igualando coeficientes,
+\begin_inset Formula 
+\begin{align*}
+B_{n-1} & =\varphi_{n}I, & B_{n-2}-CB_{n-1} & =\varphi_{n-1}I, &  & \cdots, & B_{0}-B_{1}C & =\varphi_{1}I, & -B_{0}C & =\varphi_{0}I,
+\end{align*}
+
+\end_inset
+
+y multiplicando la primera igualdad por 
+\begin_inset Formula $C^{n}$
+\end_inset
+
+, la segunda por 
+\begin_inset Formula $C^{n-1}$
+\end_inset
+
+, etc.,
+\begin_inset Formula 
+\begin{align*}
+C^{n}B_{n-1} & =\varphi_{n}C^{n}, & C^{n-1}B_{n-2}-C^{n}B_{n-1} & =\varphi_{n-1}C^{n-1}, &  & \dots,\\
+CB_{0}-C^{2}B_{1} & =\varphi_{1}C, & -CB_{0} & =\varphi_{0}I,
+\end{align*}
+
+\end_inset
+
+luego sumando es 
+\begin_inset Formula $0=\varphi_{n}C^{n}+\dots+\varphi_{1}C+\varphi_{0}=\varphi_{C}(C)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Los divisores irreducibles de 
+\begin_inset Formula $M$
+\end_inset
+
+ son precisamente los divisores irreducibles de 
+\begin_inset Formula $\varphi$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+ irreducible es divisor irreducible de 
+\begin_inset Formula $M$
+\end_inset
+
+ si y sólo si existe 
+\begin_inset Formula $v\in M\setminus\{0\}$
+\end_inset
+
+ con 
+\begin_inset Formula $pv=p(f)(v)=0$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\ker(p(f))\neq0$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $p(f):V\to V$
+\end_inset
+
+ como endomorfismo no es un isomorfismo, si y sólo si 
+\begin_inset Formula $\det(p(f))=0$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $\overline{K}$
+\end_inset
+
+ la clausura algebraica de 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $p=(X-\lambda_{1})\cdots(X-\lambda_{t})\in\overline{K}[X]$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $p\mid\varphi$
+\end_inset
+
+, sea 
+\begin_inset Formula $C$
+\end_inset
+
+ la matriz asociada a 
+\begin_inset Formula $f$
+\end_inset
+
+ bajo cualquier base, los 
+\begin_inset Formula $\lambda_{i}$
+\end_inset
+
+ son valores propios de 
+\begin_inset Formula $C$
+\end_inset
+
+ en 
+\begin_inset Formula $\overline{K}$
+\end_inset
+
+ y por tanto existen 
+\begin_inset Formula $v_{i}\in\overline{K}^{n}\setminus\{0\}$
+\end_inset
+
+ con 
+\begin_inset Formula $Cv_{i}=\lambda v_{i}$
+\end_inset
+
+ y 
+\begin_inset Formula $(C-\lambda_{i}I)=0$
+\end_inset
+
+.
+ Pero 
+\begin_inset Formula $(C-\lambda_{i}I)(C-\lambda_{j}I)=C^{2}-\lambda_{i}I-\lambda_{j}I+\lambda_{i}\lambda_{j}I=(C-\lambda_{j}I)(C-\lambda_{i}I)$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $(C-\lambda_{i}I)(C-\lambda_{j}I)(v_{i})=0$
+\end_inset
+
+ y 
+\begin_inset Formula $p(C)(v)=\left(\prod_{j}(C-\lambda_{j}I)\right)(v_{i})=0$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $\ker_{\overline{K}}(p(C))\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $\det(p(C))=0$
+\end_inset
+
+, lo que no depende de si consideramos 
+\begin_inset Formula $p(C)$
+\end_inset
+
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+ o sobre 
+\begin_inset Formula $\overline{K}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $p$
+\end_inset
+
+ es divisor irreducible de 
+\begin_inset Formula $M$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es divisor irreducible de 
+\begin_inset Formula $M$
+\end_inset
+
+, divide al mayor factor invariante de 
+\begin_inset Formula $M$
+\end_inset
+
+, 
+\begin_inset Formula $d_{t}$
+\end_inset
+
+, pero para 
+\begin_inset Formula $v\in M$
+\end_inset
+
+, 
+\begin_inset Formula $\varphi v=\varphi(f)(v)=0$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\varphi\in\text{ann}_{A}(M)=(d_{t})$
+\end_inset
+
+ y 
+\begin_inset Formula $p\mid d_{t}\mid\varphi$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{AlgL}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A,B\in{\cal M}_{n}(K)$
+\end_inset
+
+ son 
+\series bold
+semejantes
+\series default
+ si 
+\begin_inset Formula $\exists P\in{\cal M}_{n}(K):B=P^{-1}AP$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ una base de 
+\begin_inset Formula $V$
+\end_inset
+
+, 
+\begin_inset Formula $C\coloneqq M_{{\cal B}}(f)$
+\end_inset
+
+ y 
+\begin_inset Formula $f_{C}:K^{n}\to K^{n}$
+\end_inset
+
+ dado por 
+\begin_inset Formula $f_{C}(y)\coloneqq Cy$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+El isomorfismo 
+\begin_inset Formula $\phi:V\to K^{n}$
+\end_inset
+
+ que lleva 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ a la base canónica induce un isomorfismo entre el 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulo asociado a 
+\begin_inset Formula $(V,f)$
+\end_inset
+
+ y el asociado a 
+\begin_inset Formula $(K^{n},f_{C})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Claramente la biyección 
+\begin_inset Formula $\hat{\phi}$
+\end_inset
+
+ inducida conserva la suma y el producto por escalares de 
+\begin_inset Formula $K$
+\end_inset
+
+, y 
+\begin_inset Formula $\hat{\phi}(Xv)=\phi(f(v))=\phi((\phi^{-1}\circ f_{C}\circ\phi)(v))=f_{C}(\phi(v))=X\phi(v)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $W$
+\end_inset
+
+ otro 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial, 
+\begin_inset Formula $g:W\to W$
+\end_inset
+
+ un 
+\begin_inset Formula $K$
+\end_inset
+
+-endomorfismo, 
+\begin_inset Formula $\phi:V\to W$
+\end_inset
+
+ un 
+\begin_inset Formula $K$
+\end_inset
+
+-isomorfismo con 
+\begin_inset Formula $\phi\circ f=g\circ\phi$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ una base de 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+ la base correspondiente de 
+\begin_inset Formula $W$
+\end_inset
+
+ por 
+\begin_inset Formula $\phi$
+\end_inset
+
+, se tiene
+\begin_inset Formula $M_{{\cal B}}(f)=M_{{\cal B}'}(g)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula ${\cal B}\eqqcolon(b_{i})_{i}$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal B}'=(\phi(b_{i}))_{i}$
+\end_inset
+
+, pero 
+\begin_inset Formula $M_{{\cal B}}(f)$
+\end_inset
+
+ tiene como columnas los 
+\begin_inset Formula $f(b_{i})$
+\end_inset
+
+ respecto de 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ y 
+\begin_inset Formula $M_{{\cal B}'}(g)$
+\end_inset
+
+ tiene como columnas los 
+\begin_inset Formula $g(\phi(b_{i}))=\phi(f(b_{i}))$
+\end_inset
+
+ respecto de 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $M_{{\cal B}}(f)=M_{{\cal B}'}(g)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $W$
+\end_inset
+
+ es otro 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial de dimensión finita y 
+\begin_inset Formula $g:W\to W$
+\end_inset
+
+ un 
+\begin_inset Formula $K$
+\end_inset
+
+-endomorfismo, los 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulos asociados a 
+\begin_inset Formula $(V,f)$
+\end_inset
+
+ y 
+\begin_inset Formula $(W,g)$
+\end_inset
+
+ son isomorfos si y sólo si 
+\begin_inset Formula $\dim V=\dim W$
+\end_inset
+
+ y existen bases respectivas 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula $W$
+\end_inset
+
+ tales que 
+\begin_inset Formula $M_{{\cal B}}(f)$
+\end_inset
+
+ y 
+\begin_inset Formula $M_{{\cal B}'}(g)$
+\end_inset
+
+ son semejantes.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $\phi:M\to N$
+\end_inset
+
+ el isomorfismo, claramente 
+\begin_inset Formula $\phi:V\to W$
+\end_inset
+
+ es un 
+\begin_inset Formula $K$
+\end_inset
+
+-isomorfismo y por tanto 
+\begin_inset Formula $\dim_{K}V=\dim_{K}W$
+\end_inset
+
+, y basta tomar una base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ y, como 
+\begin_inset Formula $\phi(f(v))=\phi(Xv)=X\phi(v)=g(\phi(v))$
+\end_inset
+
+, estamos en las condiciones del anterior apartado.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Por cambio de base podemos suponer 
+\begin_inset Formula $M_{{\cal B}}(f)=M_{{\cal B}'}(g)\eqqcolon(a_{ij})_{1\leq i,j\leq n}$
+\end_inset
+
+, y si 
+\begin_inset Formula ${\cal B}=(b_{1},\dots,b_{n})$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}'=(b'_{1},\dots,b'_{n})$
+\end_inset
+
+, tomando el isomorfismo vectorial 
+\begin_inset Formula $\phi:V\to W$
+\end_inset
+
+ que lleva cada 
+\begin_inset Formula $b_{i}$
+\end_inset
+
+ a 
+\begin_inset Formula $b'_{i}$
+\end_inset
+
+ y viéndolo como un 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-isomorfismo 
+\begin_inset Formula $\phi:M\to N$
+\end_inset
+
+, 
+\begin_inset Formula $\phi(Xb_{i})=\phi(f(b_{i}))=\phi(\sum_{j}a_{ji}b_{j})=\sum_{j}a_{ji}b'_{j}=g(b'_{i})=X\phi(b_{i})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es una matriz cuadrada, llamamos 
+\begin_inset Formula $\text{rk}A$
+\end_inset
+
+ al rango de 
+\begin_inset Formula $A$
+\end_inset
+
+, y si 
+\begin_inset Formula $f:V\to V$
+\end_inset
+
+ es un endomorfismo, 
+\begin_inset Formula $\text{rk}f\coloneqq\text{rk}M_{{\cal B}}(f)$
+\end_inset
+
+ para cualquier base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage pagebreak
+\end_inset
+
+Llamamos 
+\series bold
+polinomio mínimo
+\series default
+ de 
+\begin_inset Formula $M$
+\end_inset
+
+ a su mayor factor invariante, elegido mónico.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $G\in K[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $j\in\mathbb{N}$
+\end_inset
+
+,
+\begin_inset Formula $\text{ann}_{M}(G^{j})=\ker(G^{j}(f))$
+\end_inset
+
+, y 
+\begin_inset Formula $G^{j}\in\text{ann}_{K[X]}(M)\iff G^{j}(f)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $G^{j}\in\text{ann}_{K[X]}(M)\iff\text{ann}_{M}(G^{j})=\ker(G^{j}(f))=M\iff G^{j}(f)=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+El polinomio mínimo de 
+\begin_inset Formula $M$
+\end_inset
+
+ es el menor 
+\begin_inset Formula $d_{t}\in K[X]$
+\end_inset
+
+ (por divisibilidad) con 
+\begin_inset Formula $d_{t}(f)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si este es 
+\begin_inset Formula $d_{t}$
+\end_inset
+
+, 
+\begin_inset Formula $(d_{t})=\text{ann}_{K[X]}(M)$
+\end_inset
+
+, y basta aplicar el apartado anterior.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es el polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+, 
+\begin_inset Formula $d_{t}\mid\varphi$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\varphi(f)=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es divisor irreducible de 
+\begin_inset Formula $M$
+\end_inset
+
+ y 
+\begin_inset Formula $n\coloneqq\min\{s\in\mathbb{N}\mid\ker(p(f)^{s})=\ker(p(f)^{s+1})\}=\min\{s\in\mathbb{N}\mid\text{rk}(p(f)^{s})=\text{rk}(p(f)^{s+1})\}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $M(p)=\ker(p(f)^{n})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\ker(p(f)^{s})=\ker(p(f)^{s+1})$
+\end_inset
+
+ implica 
+\begin_inset Formula $\text{rk}(p(f)^{s})=\text{rk}(p(f)^{s+1})$
+\end_inset
+
+, y el recíproco se cumple porque entonces 
+\begin_inset Formula $\dim\ker(p(f)^{s})=\dim\ker(p(f)^{s+1})$
+\end_inset
+
+ con 
+\begin_inset Formula $p(f)^{s}\subseteq p(f)^{s+1}$
+\end_inset
+
+.
+ Pero sabemos que 
+\begin_inset Formula $M(p)=\text{ann}_{M}(p^{n_{r}})=\ker(p(f)^{n_{r}})$
+\end_inset
+
+ siendo 
+\begin_inset Formula $n_{r}=\min\{s\in\mathbb{N}\mid\text{ann}_{M}(p^{s})=\text{ann}_{M}(p^{s+1})\}=n$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+La multiplicidad de 
+\begin_inset Formula $p$
+\end_inset
+
+ como factor irreducible de 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es 
+\begin_inset Formula $m\geq n$
+\end_inset
+
+ y cumple 
+\begin_inset Formula $M(p)=\ker(p(f)^{m})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\varphi\eqqcolon p^{m}G$
+\end_inset
+
+ con 
+\begin_inset Formula $p\nmid G$
+\end_inset
+
+, la identidad de Bézout 
+\begin_inset Formula $1=p^{m}R+GS$
+\end_inset
+
+ implica, evaluando en 
+\begin_inset Formula $f$
+\end_inset
+
+ sobre un 
+\begin_inset Formula $v\in V$
+\end_inset
+
+, que 
+\begin_inset Formula 
+\[
+v=p(f)^{m}(R(f)(v))+G(f)(S(f)(v))=R(f)(p(f)^{m}(v))+S(f)(G(f)(v)),
+\]
+
+\end_inset
+
+y por el teorema de Cayley-Hamilton, 
+\begin_inset Formula $(p^{m}G)(f)=p^{m}(f)\circ G(f)=G(f)\circ p^{m}(f)=0$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $p(f)^{m}(R(f)(v))\in\ker(G(f))$
+\end_inset
+
+ y 
+\begin_inset Formula $G(f)(S(f)(v))\in\ker(p(f)^{m})$
+\end_inset
+
+, luego 
+\begin_inset Formula $V=\ker(p(f)^{m})+\ker(G(f))$
+\end_inset
+
+ y si 
+\begin_inset Formula $v\in\ker(p(f)^{m})\cap\ker(G(f))$
+\end_inset
+
+ la igualdad anterior nos da 
+\begin_inset Formula $v=0+0=0$
+\end_inset
+
+, con lo que la suma es directa y 
+\begin_inset Formula $V=\text{ann}_{M}(p^{m})\oplus\text{ann}_{M}(G)$
+\end_inset
+
+, de donde 
+\begin_inset Formula $M(p)=\text{ann}_{M}(p^{m})=\ker(p(f)^{m})$
+\end_inset
+
+ y, por la afirmación anterior, 
+\begin_inset Formula $m\geq n$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Sea 
+\begin_inset Formula $V=V_{1}\oplus\dots\oplus V_{t}$
+\end_inset
+
+ con los 
+\begin_inset Formula $V_{i}$
+\end_inset
+
+ 
+\begin_inset Formula $f$
+\end_inset
+
+-invariantes, el polinomio mínimo de 
+\begin_inset Formula $f$
+\end_inset
+
+ es el mínimo común múltiplo de los polinomios mínimos de los 
+\begin_inset Formula $f|_{V_{i}}:V_{i}\to V_{i}$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\hat{f}_{i}\coloneqq f|_{V_{i}}:V_{i}\to V_{i}$
+\end_inset
+
+, 
+\begin_inset Formula $P$
+\end_inset
+
+ el polinomio mínimo de 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $Q_{i}$
+\end_inset
+
+ el de 
+\begin_inset Formula $\hat{f}_{i}$
+\end_inset
+
+, como 
+\begin_inset Formula $P(\hat{f}_{i})=P(f)|_{V_{i}}=0$
+\end_inset
+
+, 
+\begin_inset Formula $Q_{i}\mid P$
+\end_inset
+
+, y si 
+\begin_inset Formula $F\in K[X]$
+\end_inset
+
+ es tal que 
+\begin_inset Formula $Q_{1},\dots,Q_{t}\mid F$
+\end_inset
+
+, para 
+\begin_inset Formula $v\in V$
+\end_inset
+
+, sea 
+\begin_inset Formula $v\eqqcolon v_{1}+\dots+v_{t}$
+\end_inset
+
+ con cada 
+\begin_inset Formula $v_{i}\in V_{i}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f(v)=f(v_{1})+\dots+f(v_{t})=\hat{f}_{1}(v_{1})+\dots+\hat{f}_{t}(v_{t})=0$
+\end_inset
+
+, luego 
+\begin_inset Formula $F(f)=0$
+\end_inset
+
+ y 
+\begin_inset Formula $P\mid F$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+7.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es nilpotente, su polinomio característico es 
+\begin_inset Formula $X^{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $n\coloneqq\dim V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+8.
+\end_layout
+
+\end_inset
+
+Dados 
+\begin_inset Formula $f,g\in\text{End}_{K}V$
+\end_inset
+
+, las matrices asociadas a 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ son semejantes si y solo si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ tienen el mismo polinomio característico con factorización irreducible
+ 
+\begin_inset Formula $\varphi=p_{1}^{m_{1}}\cdots p_{k}^{m_{k}}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{rk}(p_{i}(f)^{s})=\text{rk}(p_{i}(g)^{s})$
+\end_inset
+
+ para todo 
+\begin_inset Formula $i$
+\end_inset
+
+ y 
+\begin_inset Formula $s\in\mathbb{N}^{*}$
+\end_inset
+
+, si y sólo si tienen el mismo polinomio mínimo con factorización irreducible
+ 
+\begin_inset Formula $d=p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{rk}(p_{i}(f)^{s})=\text{rk}(p_{i}(g)^{s})$
+\end_inset
+
+ para todo 
+\begin_inset Formula $i$
+\end_inset
+
+ y 
+\begin_inset Formula $s\in\mathbb{N}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Que dos endomorfismos tengan el mismo polinomio característico y el mismo
+ polinomio mínimo no implica que sus matrices asociadas bajo alguna base
+ sean semejantes.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Formas canónicas
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $F\in K[X]$
+\end_inset
+
+ mónico de grado 
+\begin_inset Formula $n>0$
+\end_inset
+
+, llamamos 
+\series bold
+matriz compañera
+\series default
+ de 
+\begin_inset Formula $F$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+C(F)\coloneqq\begin{pmatrix} &  &  & -F_{0}\\
+1 &  &  & -F_{1}\\
+ & \ddots &  & \vdots\\
+ &  & 1 & -F_{n-1}
+\end{pmatrix}\in{\cal M}_{n}(K),
+\]
+
+\end_inset
+
+y para 
+\begin_inset Formula $r>0$
+\end_inset
+
+ escribimos
+\begin_inset Formula 
+\[
+C_{r}(F)=\begin{pmatrix}\boxed{C(F)} & \boxed{U}\\
+ & \ddots & \ddots\\
+ &  & \ddots & \boxed{U}\\
+ &  &  & \boxed{C(F)}
+\end{pmatrix}\in{\cal M}_{rn}(K),
+\]
+
+\end_inset
+
+donde
+\begin_inset Formula 
+\[
+U\coloneqq\begin{pmatrix} &  & 1\\
+\\
+\\
+\end{pmatrix}\in{\cal M}_{n}(K).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El polinomio característico de un 
+\begin_inset Formula $C_{r}(F)$
+\end_inset
+
+ es 
+\begin_inset Formula $F^{r}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Primero vemos que el de 
+\begin_inset Formula $C(F)$
+\end_inset
+
+ es 
+\begin_inset Formula $F$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $n\coloneqq\text{gr}F=1$
+\end_inset
+
+, 
+\begin_inset Formula $C(F)=(-F_{0})\in{\cal M}_{1}(K)$
+\end_inset
+
+ y 
+\begin_inset Formula $\det(XI-C(F))=X+F_{0}=F$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $n>1$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{align*}
+\det(XI-C(F)) & =\begin{vmatrix}X &  &  & F_{0}\\
+-1 & \ddots &  & \vdots\\
+ & \ddots & X & F_{n-2}\\
+ &  & -1 & X+F_{n-1}
+\end{vmatrix}=\\
+ & =X\begin{vmatrix}X &  &  & F_{1}\\
+-1 & \ddots &  & \vdots\\
+ & \ddots & X & F_{n-2}\\
+ &  & -1 & X+F_{n-1}
+\end{vmatrix}+(-1)^{n+1}F_{0}\begin{vmatrix}-1 & X\\
+ & \ddots & \ddots\\
+ &  & \ddots & X\\
+ &  &  & -1
+\end{vmatrix}=\\
+ & =X(F_{1}+XF_{2}+\dots+X^{n-2}F_{n-1}+X^{n-1}F_{n})+(-1)^{n+1}(-1)^{n-1}F_{0}=F,
+\end{align*}
+
+\end_inset
+
+donde para el primer sumando hemos usado la hipótesis de inducción.
+ Para 
+\begin_inset Formula $C_{r}F$
+\end_inset
+
+, el caso 
+\begin_inset Formula $r=1$
+\end_inset
+
+ está hecho, y para 
+\begin_inset Formula $r>1$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\det(XI-C_{r}(F))=\begin{vmatrix}\boxed{C(F)} & \boxed{U}\\
+ & \ddots & \ddots\\
+ &  & \ddots & \boxed{U}\\
+ &  &  & \boxed{C(F)}
+\end{vmatrix}=\det(C(F))\det(C_{r-1}(F))=FF^{r-1}=F^{r}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+ un divisor irreducible del polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+, 
+\begin_inset Formula $h\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{v_{1},\dots,v_{t}\}\subseteq\ker(p(f)^{h})$
+\end_inset
+
+, 
+\begin_inset Formula $(\overline{v_{1}},\dots,\overline{v_{t}})$
+\end_inset
+
+ es base de 
+\begin_inset Formula $\frac{\ker(p(f)^{h})}{\ker(p(f)^{h-1})}$
+\end_inset
+
+ como 
+\begin_inset Formula $\frac{K[X]}{(p)}$
+\end_inset
+
+-espacio vectorial si y sólo si 
+\begin_inset Formula $\left(\overline{f^{i}(v_{j})}\right)_{0\leq i<d}^{1\leq j\leq t}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $\frac{\ker(p(f)^{h})}{\ker(p(f)^{h-1})}$
+\end_inset
+
+ como 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial.
+ En particular, si 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+ es mónico irreducible con 
+\begin_inset Formula $p(f)=0$
+\end_inset
+
+ y 
+\begin_inset Formula $\{v_{1},\dots,v_{t}\}\subseteq V$
+\end_inset
+
+, 
+\begin_inset Formula $(v_{1},\dots,v_{t})$
+\end_inset
+
+ es base de 
+\begin_inset Formula $M$
+\end_inset
+
+ como 
+\begin_inset Formula $\frac{K[X]}{(p)}$
+\end_inset
+
+-espacio vectorial si y sólo si 
+\begin_inset Formula $(f^{i}(v_{j}))_{0\leq i<d}^{1\leq j\leq t}$
+\end_inset
+
+ es base del 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $F\in K[X]$
+\end_inset
+
+ un polinomio mónico de grado 
+\begin_inset Formula $n>0$
+\end_inset
+
+ y 
+\begin_inset Formula $r\in\mathbb{N}^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $M\cong\frac{K[X]}{(F^{r})}$
+\end_inset
+
+ si y sólo si existe 
+\begin_inset Formula $v\in V$
+\end_inset
+
+ tal que 
+\begin_inset Formula $(f^{s}(v))_{s=0}^{rn-1}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $v$
+\end_inset
+
+ y 
+\begin_inset Formula $F(f)^{r}(v)=0$
+\end_inset
+
+, si y sólo si existe una base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ con 
+\begin_inset Formula $M_{{\cal B}}(f)=C_{r}(F)$
+\end_inset
+
+, en cuyo caso el polinomio mínimo de 
+\begin_inset Formula $M$
+\end_inset
+
+ coincide con el polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+ y es 
+\begin_inset Formula $F^{r}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies3]$
+\end_inset
+
+ Sea 
+\begin_inset Formula ${\cal \tilde{B}}_{j}\coloneqq(\overline{F^{j}},\overline{XF^{j}},\dots,\overline{X^{n-1}F^{j}})$
+\end_inset
+
+ para 
+\begin_inset Formula $j\in\{0,\dots,r-1\}$
+\end_inset
+
+ y 
+\begin_inset Quotes cld
+\end_inset
+
+
+\begin_inset Formula $\star$
+\end_inset
+
+
+\begin_inset Quotes crd
+\end_inset
+
+ la concatenación de secuencias, 
+\begin_inset Formula $\tilde{{\cal B}}\coloneqq\tilde{{\cal B}}_{r-1}\star\dots\star\tilde{{\cal B}}_{1}\star\tilde{{\cal B}}_{0}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $\frac{K[X]}{(F^{r})}$
+\end_inset
+
+ como 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial.
+ Para verlo, como 
+\begin_inset Formula $|\tilde{{\cal B}}|=rn=\dim\frac{K[X]}{(F^{r})}$
+\end_inset
+
+, basta ver que 
+\begin_inset Formula $\tilde{{\cal B}}$
+\end_inset
+
+ es linealmente independiente.
+ Si 
+\begin_inset Formula $r=1$
+\end_inset
+
+, 
+\begin_inset Formula $\tilde{{\cal B}}=(\overline{1},\overline{X},\dots,\overline{X}^{n-1})$
+\end_inset
+
+ y el resultado es claro.
+ Si 
+\begin_inset Formula $r>1$
+\end_inset
+
+, sea 
+\begin_inset Formula $\sum_{i=0}^{n-1}\sum_{j=0}^{r-1}\lambda_{ij}X^{i}F^{j}=0\in\frac{K[X]}{(F^{r})}$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $\lambda_{ij}\in K$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\sum_{ij}\lambda_{ij}X^{i}F^{j}=F^{r}G\in K[X]$
+\end_inset
+
+ para cierto 
+\begin_inset Formula $G\in K[X]$
+\end_inset
+
+, pero 
+\begin_inset Formula $\sum_{ij}\lambda_{ij}X^{i}F^{j}=\sum_{i=0}^{n-1}\lambda_{i0}X^{i}+F(\sum_{i=0}^{n-1}\sum_{j=1}^{r-1}\lambda_{ij}X^{i}F^{j})$
+\end_inset
+
+, luego debe ser 
+\begin_inset Formula $F\mid\sum_{i=0}^{n-1}\lambda_{i0}X^{i}$
+\end_inset
+
+ y, como 
+\begin_inset Formula $\text{gr}F=n$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{i=0}^{n-1}\lambda_{i0}X^{i}=0$
+\end_inset
+
+ y cada 
+\begin_inset Formula $\lambda_{i0}=0$
+\end_inset
+
+.
+ Pero entonces, dividiendo por 
+\begin_inset Formula $F$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{i=0}^{n-1}\sum_{j=1}^{r-1}\lambda_{ij}X^{i}F^{j-1}=F^{r-1}G$
+\end_inset
+
+ y por hipótesis de inducción todos los 
+\begin_inset Formula $\lambda_{ij}=0$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $g:\frac{K[X]}{(F^{r})}\to\frac{K[X]}{(F^{r})}$
+\end_inset
+
+ el endomorfismo 
+\begin_inset Formula $G\mapsto XG$
+\end_inset
+
+, queremos ver que 
+\begin_inset Formula $C\coloneqq M_{{\cal B}}(g)=C_{r}(F)$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $j\in\{0,\dots,r-1\}$
+\end_inset
+
+, 
+\begin_inset Formula $g(\tilde{{\cal B}}_{j})=(\overline{XF^{j}},\overline{X^{2}F^{j}},\dots,\overline{X^{n}F^{j}})$
+\end_inset
+
+, pero
+\begin_inset Formula 
+\[
+\overline{F^{j+1}}-\overline{X^{n}F^{j}}=\overline{(F-X^{n})F^{j}}=\left(\sum_{i=0}^{n-1}F_{i}\overline{X^{i}}\right)\overline{F^{j}}=\sum_{i=0}^{n-1}F_{i}\overline{X^{i}F^{j}}
+\]
+
+\end_inset
+
+y por tanto
+\begin_inset Formula 
+\[
+\overline{X^{n}F^{j}}=\overline{F^{j+1}}-\sum_{i=0}^{n-1}F_{i}\overline{X^{i}F^{j}}.
+\]
+
+\end_inset
+
+Entonces, para 
+\begin_inset Formula $j=r-1$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{F^{r+1}}=0$
+\end_inset
+
+ y las primeras 
+\begin_inset Formula $n$
+\end_inset
+
+ columnas de 
+\begin_inset Formula $C$
+\end_inset
+
+ solo tienen entradas no nulas en las primeras 
+\begin_inset Formula $n$
+\end_inset
+
+ filas y estas entradas son
+\begin_inset Formula 
+\[
+\begin{pmatrix} &  &  & -F_{0}\\
+1 &  &  & -F_{1}\\
+ & \ddots &  & \vdots\\
+ &  & 1 & -F_{n-1}
+\end{pmatrix}=C(F),
+\]
+
+\end_inset
+
+mientras que para 
+\begin_inset Formula $j<r-1$
+\end_inset
+
+, 
+\begin_inset Formula $\overline{F^{j+1}}$
+\end_inset
+
+ es un elemento de la base y las columnas de 
+\begin_inset Formula $C$
+\end_inset
+
+ correspondientes a 
+\begin_inset Formula $\tilde{{\cal B}}_{j}$
+\end_inset
+
+ solo tienen entradas no nulas en las filas de 
+\begin_inset Formula $\tilde{{\cal B}}_{j}$
+\end_inset
+
+, formando la submatriz 
+\begin_inset Formula $C(F)$
+\end_inset
+
+, y en la columna de 
+\begin_inset Formula $\overline{X^{n-1}F^{j}}$
+\end_inset
+
+ con la fila de 
+\begin_inset Formula $\overline{F^{j+1}}$
+\end_inset
+
+, dando la submatriz 
+\begin_inset Formula $U$
+\end_inset
+
+ de la definición de 
+\begin_inset Formula $C_{r}(F)$
+\end_inset
+
+.
+ Finalmente, el 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-módulo generado por 
+\begin_inset Formula $(\frac{K[X]}{(F^{r})},g)$
+\end_inset
+
+ es claramente 
+\begin_inset Formula $\frac{K[X]}{(F^{r})}$
+\end_inset
+
+, y si 
+\begin_inset Formula $\phi:M\to\frac{K[X]}{(F^{r})}$
+\end_inset
+
+ es el isomorfismo de la hipótesis, como 
+\begin_inset Formula $\phi(f(v))=\phi(Xv)=X\phi(v)=g(\phi(v))$
+\end_inset
+
+, tomando la base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ inducida por 
+\begin_inset Formula $\tilde{{\cal B}}$
+\end_inset
+
+ mediante 
+\begin_inset Formula $\phi^{-1}$
+\end_inset
+
+ queda 
+\begin_inset Formula $M_{{\cal B}}(f)=M_{\tilde{{\cal B}}}(g)=C_{r}(F)$
+\end_inset
+
+, y el polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+ es el de 
+\begin_inset Formula $C_{r}(F)$
+\end_inset
+
+ que es 
+\begin_inset Formula $F^{r}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ Tomando 
+\begin_inset Formula $g$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{{\cal B}}$
+\end_inset
+
+ de la parte anterior de la prueba, 
+\begin_inset Formula $M_{{\cal B}}(f)=C_{r}(f)=M_{\tilde{B}}(g)$
+\end_inset
+
+ y, como esto también significa que 
+\begin_inset Formula $\dim V=\dim\frac{K[X]}{(F^{r})}$
+\end_inset
+
+, queda el isomorfismo 
+\begin_inset Formula $M\to\frac{K[X]}{(F^{r})}$
+\end_inset
+
+ deseado, y como 
+\begin_inset Formula $\text{ann}_{K[X]}(M)=\text{ann}_{K[X]}\frac{K[X]}{(F^{r})}=(F^{r})$
+\end_inset
+
+ y 
+\begin_inset Formula $F^{r}$
+\end_inset
+
+ es mónico, 
+\begin_inset Formula $F^{r}$
+\end_inset
+
+ es el polinomio mínimo de 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies2]$
+\end_inset
+
+ Sea 
+\begin_inset Formula $\phi:\frac{K[X]}{(F^{r})}\to M$
+\end_inset
+
+ un 
+\begin_inset Formula $K[X]$
+\end_inset
+
+-isomorfismo, que induce un 
+\begin_inset Formula $K$
+\end_inset
+
+-isomorfismo 
+\begin_inset Formula $\phi:\frac{K[X]}{(F^{r})}\to V$
+\end_inset
+
+, como 
+\begin_inset Formula $(\overline{1},\overline{X},\dots,\overline{X}^{rn-1})$
+\end_inset
+
+ es base de 
+\begin_inset Formula $\frac{K[X]}{(F^{r})}$
+\end_inset
+
+, tomando 
+\begin_inset Formula $v\coloneqq\phi(\overline{1})$
+\end_inset
+
+, 
+\begin_inset Formula $(\overline{v},\overline{f(v)},\dots,\overline{f^{rn-1}(v)})$
+\end_inset
+
+ es base de 
+\begin_inset Formula $V$
+\end_inset
+
+ y 
+\begin_inset Formula $F(f)^{r}(v)=F^{r}(f)(v)=\overline{F^{r}}=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies1]$
+\end_inset
+
+ Para 
+\begin_inset Formula $w\in M=V$
+\end_inset
+
+, existen 
+\begin_inset Formula $b_{s}\in K$
+\end_inset
+
+ con 
+\begin_inset Formula $w=\sum_{s=0}^{rn-1}b_{s}f^{s}(v)=(\sum_{s=0}^{rn-1}b_{s}X^{s})v$
+\end_inset
+
+, luego 
+\begin_inset Formula $M=(v)$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi:K[X]\twoheadrightarrow M$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\pi(G)\coloneqq Gv$
+\end_inset
+
+ es un epimorfismo, pero 
+\begin_inset Formula $F^{r}\in\ker\pi$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\pi$
+\end_inset
+
+ induce un epimorfismo 
+\begin_inset Formula $\hat{\pi}:\frac{K[X]}{(F^{r})}\twoheadrightarrow M$
+\end_inset
+
+, y como 
+\begin_inset Formula $\dim_{K}\frac{K[X]}{(F^{r})}=rn=\dim_{K}M$
+\end_inset
+
+, 
+\begin_inset Formula $\hat{\pi}$
+\end_inset
+
+ es un isomorfismo.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de clasificación de endomorfismos:
+\series default
+ Existen una base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+, 
+\begin_inset Formula $h_{1},\dots,h_{t}\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $p_{1},\dots,p_{t}\in K[X]$
+\end_inset
+
+ irreducibles tales que
+\begin_inset Formula 
+\[
+M_{{\cal B}}(f)=\begin{pmatrix}\boxed{C_{h_{1}}(p_{1})}\\
+ & \ddots\\
+ &  & \boxed{C_{h_{t}}(p_{t})}
+\end{pmatrix},
+\]
+
+\end_inset
+
+siendo esta matriz, llamada 
+\series bold
+forma canónica
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+, unívocamente determinada por 
+\begin_inset Formula $f$
+\end_inset
+
+ salvo reordenación de bloques y formada, exactamente, por
+\begin_inset Formula 
+\[
+\frac{\text{rk}(p(f)^{h-1})+\text{rk}(p(f)^{h+1})-2\text{rk}(p(f)^{h})}{\text{rg}p}
+\]
+
+\end_inset
+
+bloques 
+\begin_inset Formula $C_{h}(p)$
+\end_inset
+
+ para cada divisor irreducible mónico 
+\begin_inset Formula $p$
+\end_inset
+
+ del polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+ y cada 
+\begin_inset Formula $h\leq\min\{s\in\mathbb{N}^{*}\mid\text{rk}(p(f)^{s})=\text{rk}(p(f)^{s+1})\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $M=\bigoplus_{i=1}^{k}\bigoplus_{j=1}^{r_{i}}N_{ij}$
+\end_inset
+
+ una descomposición canónica con cada 
+\begin_inset Formula $N_{ij}\cong\frac{K[X]}{(p_{i}^{n_{ij}})}$
+\end_inset
+
+, cada 
+\begin_inset Formula $N_{ij}$
+\end_inset
+
+ es un subespacio 
+\begin_inset Formula $f$
+\end_inset
+
+-invariante de 
+\begin_inset Formula $V$
+\end_inset
+
+, por lo que existe una base 
+\begin_inset Formula ${\cal B}_{ij}$
+\end_inset
+
+ de 
+\begin_inset Formula $N_{ij}$
+\end_inset
+
+ como 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial con 
+\begin_inset Formula $M_{{\cal B}_{ij}}(f|_{N_{ij}})=C_{n_{ij}}(p_{i})$
+\end_inset
+
+, y uniendo las bases se obtiene una base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ con 
+\begin_inset Formula $M_{{\cal B}}(f)$
+\end_inset
+
+ de la forma buscada.
+\end_layout
+
+\begin_layout Standard
+Si ahora 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+ es otra base tal que 
+\begin_inset Formula $M_{{\cal B}}(f)$
+\end_inset
+
+ está formada por bloques diagonales 
+\begin_inset Formula $(C_{h_{s}}(q_{s}))_{s=1}^{u}$
+\end_inset
+
+, 
+\begin_inset Formula $V$
+\end_inset
+
+ se puede descomponer en suma directa interna de subespacios 
+\begin_inset Formula $f$
+\end_inset
+
+-invariantes 
+\begin_inset Formula $W_{s}$
+\end_inset
+
+ con bases 
+\begin_inset Formula ${\cal B}_{s}$
+\end_inset
+
+ tales que, si 
+\begin_inset Formula $\hat{f}_{s}\coloneqq f|_{W_{s}}:W_{s}\to W_{S}$
+\end_inset
+
+, 
+\begin_inset Formula $M_{{\cal B}_{s}}(\hat{f}_{s})=C_{h_{s}}(q_{s})$
+\end_inset
+
+, con lo que el módulo generado por 
+\begin_inset Formula $(W_{s},\hat{f}_{s})$
+\end_inset
+
+ es un submódulo no nulo de 
+\begin_inset Formula $M$
+\end_inset
+
+ isomorfo a 
+\begin_inset Formula $\frac{K[X]}{(q_{s}^{h_{s}})}$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $M=\bigoplus_{s=1}^{u}\frac{K[X]}{(q_{s}^{h_{s}})}$
+\end_inset
+
+ y, como las descomposiciones de esta forma son únicas, los bloques son
+ los mismos que en la descomposición que hemos encontrado y los irreducibles
+ que aparecen son los divisores irreducibles de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para la última parte, otra forma de obtener la forma canónica de cada 
+\begin_inset Formula $M(p)$
+\end_inset
+
+ es usando los 
+\begin_inset Formula $(F_{h})_{h=1}^{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $n\coloneqq\max_{i}r_{i}=\min\{s\in\mathbb{N}^{*}\mid\text{ann}_{M(p)}(p^{s})=\text{ann}_{M(p)}(p^{s+1})\}$
+\end_inset
+
+, cada 
+\begin_inset Formula $F_{h}\subseteq\text{ann}_{M}(p^{h})$
+\end_inset
+
+ y tales que cada 
+\begin_inset Formula $F_{h}\dot{\cup}pF_{h+1}\dot{\cup}\dots\dot{\cup}p^{n-h}F_{n}$
+\end_inset
+
+ induce una base de 
+\begin_inset Formula $\frac{\text{ann}_{M}(p^{h})}{\text{ann}_{M}(p^{h-1})}=\frac{\ker(p(f)^{h})}{\ker(p(f)^{h-1})}$
+\end_inset
+
+ como 
+\begin_inset Formula $\frac{K[X]}{(p)}$
+\end_inset
+
+-espacio vectorial.
+ Si 
+\begin_inset Formula $\hat{f}\coloneqq f|_{M(p)}:M(p)\to M(p)$
+\end_inset
+
+, 
+\begin_inset Formula $\text{ann}_{M(p)}(p^{s})=\ker(p(\hat{f})^{s})=\ker(p(f)^{s})$
+\end_inset
+
+ ya que 
+\begin_inset Formula $p(f)^{s}(v)=0\implies p^{s}v=0\implies v\in\text{ann}_{M}(p^{s})\subseteq M(p)$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $n=\min\{s\in\mathbb{N}^{*}\mid\text{rk}(p(f)^{s})=\text{rk}(p(f)^{s+1})\}$
+\end_inset
+
+.
+ Además, el número de apariciones de 
+\begin_inset Formula $p^{s}$
+\end_inset
+
+ como divisor elemental de 
+\begin_inset Formula $M$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu_{h}=\delta_{h}-\delta_{h+1}\coloneqq\dim_{\frac{K[X]}{(p)}}\frac{\ker(p(f)^{h})}{\ker(p(f)^{h-1})}-\dim_{\frac{K[X]}{(p)}}\frac{\ker(p(f)^{h+1})}{\ker(p(f)^{h})}$
+\end_inset
+
+, pero es fácil ver que todo 
+\begin_inset Formula $\frac{K[X]}{(p)}$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $U$
+\end_inset
+
+ es un 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial y 
+\begin_inset Formula $\dim_{\frac{K[X]}{(p)}}(U)=\frac{\dim_{K}(U)}{\text{gr}p}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\mu_{h}=\frac{1}{\text{gr}p}(\dim_{K}\ker(p(f)^{h})-\dim_{K}\ker(p(f)^{h-1})-\dim_{K}\ker(p(f)^{h+1})+\dim_{K}\ker(p(f)^{h}))$
+\end_inset
+
+ y el resultado sale de que 
+\begin_inset Formula $\dim_{K}\ker(p(f)^{h})=\dim_{K}V-\text{rk}(p(f)^{h})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, toda 
+\begin_inset Formula $C\in{\cal M}_{n}(K)$
+\end_inset
+
+ es semejante a una de la forma
+\begin_inset Formula 
+\[
+\begin{pmatrix}\boxed{C_{h_{1}}(p_{1})}\\
+ & \ddots\\
+ &  & \boxed{C_{h_{t}}(p_{t})}
+\end{pmatrix}
+\]
+
+\end_inset
+
+con los 
+\begin_inset Formula $p_{i}\in K[X]$
+\end_inset
+
+ irreducibles, siendo esta matriz, llamada 
+\series bold
+forma canónica
+\series default
+ de 
+\begin_inset Formula $C$
+\end_inset
+
+, unívocamente determinada por 
+\begin_inset Formula $C$
+\end_inset
+
+ salvo reordenación de bloques y formada, exactamente, por
+\begin_inset Formula 
+\[
+\frac{\text{rk}(p(C)^{h-1})+\text{rk}(p(C)^{h+1})-2\text{rk}(p(C)^{h})}{\text{rg}p}
+\]
+
+\end_inset
+
+bloques 
+\begin_inset Formula $C_{h}(p)$
+\end_inset
+
+ para cada divisor irreducible mónico 
+\begin_inset Formula $p$
+\end_inset
+
+ del polinomio característico de 
+\begin_inset Formula $p$
+\end_inset
+
+ y cada 
+\begin_inset Formula $h\leq\min\{s\in\mathbb{N}^{*}\mid\text{rk}(p(f)^{s})=\text{rk}(p(f)^{s+1})\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $F\in K[X]$
+\end_inset
+
+ es no constante con factorización irreducible 
+\begin_inset Formula $F=p_{1}^{m_{1}}\cdots p_{k}^{m_{k}}$
+\end_inset
+
+ con los 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ mónicos irreducibles distintos, la forma canónica de la matriz compañera
+ 
+\begin_inset Formula $C$
+\end_inset
+
+ de 
+\begin_inset Formula $F$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\begin{pmatrix}\boxed{C_{m_{1}}(p_{1})}\\
+ & \ddots\\
+ &  & \boxed{C_{m_{k}}(p_{k})}
+\end{pmatrix},
+\]
+
+\end_inset
+
+y en particular 
+\begin_inset Formula $C$
+\end_inset
+
+ tiene un único divisor elemental asociado a cada divisor mónico irreducible
+ de 
+\begin_inset Formula $F$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Formas de Jordan
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+valor propio
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ es un 
+\begin_inset Formula $\lambda\in K$
+\end_inset
+
+ tal que 
+\begin_inset Formula $X-\lambda$
+\end_inset
+
+ divide al polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+, y su 
+\series bold
+multiplicidad geométrica
+\series default
+ es 
+\begin_inset Formula $\nu_{\text{g}}(\lambda)\coloneqq\dim_{K}\ker(f-\lambda1_{V})>0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{exinfo}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $\lambda\in K$
+\end_inset
+
+, 
+\begin_inset Formula $C(X-\lambda)=(\lambda)\in{\cal M}_{1}(K)$
+\end_inset
+
+ y, para 
+\begin_inset Formula $r>0$
+\end_inset
+
+, llamamos 
+\series bold
+bloque de Jordan
+\series default
+ de tamaño 
+\begin_inset Formula $r$
+\end_inset
+
+ asociado al valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ a 
+\begin_inset Formula $J_{r}(\lambda)\coloneqq C_{r}(X-\lambda)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Jordan:
+\end_layout
+
+\begin_layout Enumerate
+Si el polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+ se descompone completamente en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+, existe una base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ tal que 
+\begin_inset Formula 
+\[
+M_{{\cal B}}(f)=\begin{pmatrix}\boxed{J_{h_{1}}(\lambda_{1})}\\
+ & \ddots\\
+ &  & \boxed{J_{h_{t}}(\lambda_{t})}
+\end{pmatrix}
+\]
+
+\end_inset
+
+para ciertos 
+\begin_inset Formula $h_{i}>0$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda_{i}\in K$
+\end_inset
+
+, siendo esta matriz unívocamente determinada por 
+\begin_inset Formula $f$
+\end_inset
+
+ salvo reordenación de bloques y formada por 
+\begin_inset Formula $\text{rk}((f-\lambda1_{V})^{h-1})+\text{rk}((f-\lambda1_{V})^{h+1})-2\text{rk}((f-\lambda1_{V})^{h})$
+\end_inset
+
+ bloques 
+\begin_inset Formula $J_{h}(\lambda)$
+\end_inset
+
+ para cada valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ y cada 
+\begin_inset Formula $h\in\mathbb{N}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Por el teorema de clasificación de endomorfismos usando que los irreducibles
+ del polinomio característico son los 
+\begin_inset Formula $X-\lambda$
+\end_inset
+
+ con 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ valor propio de 
+\begin_inset Formula $f$
+\end_inset
+
+ y que el grado de estos es 1.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $C\in{\cal M}_{n}(K)$
+\end_inset
+
+ es una matriz cuadrada cuyo polinomio característico se descompone completament
+e en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+, 
+\begin_inset Formula $C$
+\end_inset
+
+ es semejante a una matriz como la del apartado anterior, única salvo reordenaci
+ón de bloques y formada por 
+\begin_inset Formula $\text{rk}((C-\lambda I)^{h-1})+\text{rk}((C-\lambda I)^{h+1})-2\text{rk}((C-\lambda I)^{h})$
+\end_inset
+
+ bloques 
+\begin_inset Formula $J_{h}(\lambda)$
+\end_inset
+
+ para cada valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ de 
+\begin_inset Formula $C$
+\end_inset
+
+ y cada 
+\begin_inset Formula $h\in\mathbb{N}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ el polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $p$
+\end_inset
+
+ un divisor mónico irreducible de grado 
+\begin_inset Formula $d$
+\end_inset
+
+ y multiplicidad 1:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $M(p)=\ker(p(f))\cong\frac{K[X]}{(p)}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Claramente 
+\begin_inset Formula $\ker(p(f))\subseteq M(p)$
+\end_inset
+
+, y si 
+\begin_inset Formula $x\in M(p)$
+\end_inset
+
+, existe 
+\begin_inset Formula $s>0$
+\end_inset
+
+ con 
+\begin_inset Formula $p^{s}x=0$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in\ker(p(f)^{s})$
+\end_inset
+
+, pero como la multiplicidad de 
+\begin_inset Formula $p$
+\end_inset
+
+ en 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es 1, 
+\begin_inset Formula $\ker(p(f))=\ker(p(f)^{s})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para todo 
+\begin_inset Formula $v\in M(p)\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal B}\coloneqq\{f^{s}(v)\}_{s\in\mathbb{N}_{d}}$
+\end_inset
+
+ es una base de 
+\begin_inset Formula $\ker(p(f))$
+\end_inset
+
+ y 
+\begin_inset Formula $M_{{\cal B}}(f|_{M(p)}:M(p)\to M(p))=C_{1}(p)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\phi_{0}:\frac{K[X]}{(p)}\to M(p)$
+\end_inset
+
+ un isomorfismo, 
+\begin_inset Formula $\overline{q}\coloneqq(\phi_{0})^{-1}(v)\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi:\frac{K[X]}{(p)}\twoheadrightarrow\frac{K[X]}{(p)}$
+\end_inset
+
+ el epimorfismo 
+\begin_inset Formula $\pi(\overline{F})\coloneqq\overline{qF}$
+\end_inset
+
+, como 
+\begin_inset Formula $\gcd\{p,q\}=1$
+\end_inset
+
+, existe una identidad de Bézout 
+\begin_inset Formula $1=pR+qS$
+\end_inset
+
+, luego 
+\begin_inset Formula $\overline{1}=\overline{qS}\in\text{Im}\pi$
+\end_inset
+
+ y 
+\begin_inset Formula $\pi$
+\end_inset
+
+ es un isomorfismo.
+ Por tanto 
+\begin_inset Formula $\phi\coloneqq\phi_{0}\circ\pi L\frac{K[X]}{(p)}\to M(p)$
+\end_inset
+
+ es un isomorfismo con 
+\begin_inset Formula $\phi(\overline{1})=v$
+\end_inset
+
+ y, como 
+\begin_inset Formula $(X^{s})_{s\in\mathbb{N}_{d}}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $\frac{K[X]}{(p)}$
+\end_inset
+
+ como 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial, 
+\begin_inset Formula ${\cal B}\coloneqq(f^{s}(v))_{s\in\mathbb{N}_{d}}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $M(p)$
+\end_inset
+
+ como 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial.
+ Ahora bien, si 
+\begin_inset Formula $b_{i}\coloneqq f^{i}(v)$
+\end_inset
+
+, para 
+\begin_inset Formula $i\in\{0,\dots,d-2\}$
+\end_inset
+
+, 
+\begin_inset Formula $f(b_{i})=f(f^{i}(v))=f^{i+1}(v)=b_{i+1}$
+\end_inset
+
+, y para 
+\begin_inset Formula $d-1$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+f(b_{d-1})=f^{d}(v)=\phi(X^{d})=\phi(X^{d}-p)=\phi\left(-\sum_{i=0}^{d-1}p_{i}X^{i}\right)=\sum_{i=0}^{d-1}-p_{i}b_{i},
+\]
+
+\end_inset
+
+lo que nos da 
+\begin_inset Formula $M_{{\cal B}}(f)=C(p)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Análogamente, si 
+\begin_inset Formula $C\in{\cal M}_{n}(K)$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+ es un irreducible con multiplicidad 1 en el polinomio característico de
+ 
+\begin_inset Formula $C$
+\end_inset
+
+, la forma canónica de 
+\begin_inset Formula $C$
+\end_inset
+
+ tiene exactamente un bloque de la forma 
+\begin_inset Formula $C_{h}(p)$
+\end_inset
+
+ que es precisamente 
+\begin_inset Formula $C(p)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un 
+\begin_inset Formula $\lambda\in\mathbb{R}$
+\end_inset
+
+ es un 
+\series bold
+valor propio simple
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ o de 
+\begin_inset Formula $C\in{\cal M}_{n}(K)$
+\end_inset
+
+ si 
+\begin_inset Formula $X-\lambda$
+\end_inset
+
+ es divisor de su polinomio característico con multiplicidad 1, en cuyo
+ caso:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $M(X-\lambda)=\ker((X-\lambda)(f))=\{v\in V\mid f(v)=\lambda v\}\cong\frac{K[X]}{(X-\lambda)}$
+\end_inset
+
+ es el subespacio propio de 
+\begin_inset Formula $V$
+\end_inset
+
+ asociado al valor propio 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para todo 
+\begin_inset Formula $v\in M(X-\lambda)\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $M(X-\lambda)=(v)$
+\end_inset
+
+ y 
+\begin_inset Formula $f|_{(v)}$
+\end_inset
+
+ es el producto por 
+\begin_inset Formula $\lambda$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La forma canónica de 
+\begin_inset Formula $C$
+\end_inset
+
+ tiene un único bloque de la forma 
+\begin_inset Formula $J_{h}(\lambda)$
+\end_inset
+
+, que es 
+\begin_inset Formula $J(\lambda)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Anillos de polinomios y matrices
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $B\in\text{GL}_{s}(K)$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+C\coloneqq\begin{pmatrix} & \boxed{B} & \boxed{I_{s}}\\
+ &  & \ddots & \ddots\\
+ &  &  & \ddots & \boxed{I_{s}}\\
+ &  &  &  & \boxed{B}\\
+\\
+\end{pmatrix}\in{\cal M}_{rs}(K),
+\]
+
+\end_inset
+
+para 
+\begin_inset Formula $k\in\{1,\dots,r-1\}$
+\end_inset
+
+, viendo 
+\begin_inset Formula $C^{k}$
+\end_inset
+
+ por bloques como elemento de 
+\begin_inset Formula ${\cal M}_{r}({\cal M}_{s}(K))$
+\end_inset
+
+, su 
+\begin_inset Formula $k$
+\end_inset
+
+-ésima diagonal por encima de la principal está formada por copias de 
+\begin_inset Formula $B^{k}$
+\end_inset
+
+ y las de debajo de dicha diagonal son nulas, y 
+\begin_inset Formula $C^{r}=0\neq C^{r-1}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $\phi:{\cal M}_{rs}(K)\to{\cal M}_{r}({\cal M}_{s}(K))$
+\end_inset
+
+ que agrupa las matrices en bloques es un isomorfismo de anillos, pues clarament
+e conserva la suma y la identidad y, para el producto, haciendo los índices
+ de matrices empezar por 0 por simplicidad,
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Como debería ser siempre.
+\end_layout
+
+\end_inset
+
+ si 
+\begin_inset Formula $A,B\in{\cal M}_{rs}(K)$
+\end_inset
+
+, para 
+\begin_inset Formula $i,j\in\{0,\dots,r-1\}$
+\end_inset
+
+ y 
+\begin_inset Formula $k,l\in\{1,\dots,s\}$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{align*}
+(\phi(A)\phi(B))_{ijkl} & =\left(\sum_{p\in\mathbb{N}_{r}}\phi(A)_{ir}\phi(B)_{rj}\right)_{kl}=\sum_{p\in\mathbb{N}_{r}}\left(\phi(A)_{ip}\phi(B)_{pj}\right)_{kl}=\\
+ & =\sum_{p\in\mathbb{N}_{r}}\sum_{q\in\mathbb{N}_{s}}\phi(A)_{ipkq}\phi(B)_{pjql}=\sum_{p\in\mathbb{N}_{r}}\sum_{q\in\mathbb{N}_{s}}A_{is+k,ps+q}B_{ps+q,js+l}=\\
+ & =\sum_{z\in\mathbb{N}_{rs}}A_{is+k,z}B_{z,js+l}=(AB)_{is+k,js+l}=\phi(AB)_{ijkl}.
+\end{align*}
+
+\end_inset
+
+Entonces, si 
+\begin_inset Formula $C\in{\cal M}_{r}({\cal M}_{s}(K))$
+\end_inset
+
+, queremos ver que cada 
+\begin_inset Formula $(C^{k})_{ij}=\binom{k}{2k+i-j}B^{2k+i-j}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $(C^{k})_{i,i+k}=\binom{k}{k}B^{k}=B^{k}$
+\end_inset
+
+ y, para 
+\begin_inset Formula $j<i+k$
+\end_inset
+
+, 
+\begin_inset Formula $2k+i-j>k$
+\end_inset
+
+ y 
+\begin_inset Formula $\binom{k}{2k+i-j}=0$
+\end_inset
+
+.
+ Por inducción, para 
+\begin_inset Formula $k=1$
+\end_inset
+
+, 
+\begin_inset Formula $C_{i,i+1}=B=\binom{1}{1}B^{1}$
+\end_inset
+
+, 
+\begin_inset Formula $C_{i,i+2}=I_{s}=\binom{1}{0}B^{0}$
+\end_inset
+
+ y el resto de entradas son nulas, y para 
+\begin_inset Formula $k>1$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{align*}
+(C^{k})_{ij} & =\sum_{l=1}^{r}(C^{k-1})_{il}C_{lj}=\sum_{l=1}^{r}\binom{k-1}{2k-2+i-l}\binom{1}{2+l-j}B^{2k-2+i-l+2-j+l}=\\
+ & =\sum_{l}\binom{k-1}{(1-k-i)+l}\binom{1}{(2-j)+l}B^{2k+i-j}=\binom{k}{2k+i-j}B^{2k+i-j},
+\end{align*}
+
+\end_inset
+
+donde en la última igualdad hemos usado que 
+\begin_inset Formula $\sum_{k}\binom{r}{m+k}\binom{s}{n+k}=\binom{r+s}{r-m+n}$
+\end_inset
+
+ y en la penúltima hemos usado que 
+\begin_inset Formula $(k-1)-(2k-2+i-l)=1-k-i+l$
+\end_inset
+
+ y que podemos expandir el rango del sumatorio ya que, si el producto de
+ los dos coeficientes no se anula, entonces 
+\begin_inset Formula $2+l-j\in\{0,1\}\implies l\leq j-1<r$
+\end_inset
+
+ y 
+\begin_inset Formula $0\leq1-k-i+l\leq k-1\implies k-1\leq l-i\leq2(k-1)\implies l\geq k+i-1>1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $C\in{\cal M}_{n}(K)$
+\end_inset
+
+, 
+\begin_inset Formula $P\in\text{GL}_{n}(K)$
+\end_inset
+
+ y 
+\begin_inset Formula $C'\coloneqq PCP^{-1}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $F\in K[X]$
+\end_inset
+
+, 
+\begin_inset Formula $F(C')=PF(C)P^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $(PCP^{-1})^{k}=PC^{k}P^{-1}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $F(PCP^{-1})=\sum_{k}F_{k}PC^{k}P^{-1}\overset{F_{k}\in K}{=}P(\sum_{k}F_{k}C^{k})P^{-1}=PF(C)P^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $C$
+\end_inset
+
+ y 
+\begin_inset Formula $C'$
+\end_inset
+
+ tienen el mismo polinomio mínimo.
+ 
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Por lo anterior, usando que el polinomio mínimo de una matriz 
+\begin_inset Formula $C$
+\end_inset
+
+ es el menor 
+\begin_inset Formula $d_{t}$
+\end_inset
+
+ con 
+\begin_inset Formula $d_{t}(C)=0$
+\end_inset
+
+ y que 
+\begin_inset Formula $F(C')=PF(C)P^{-1}=0\iff F(C)=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Formas canónicas reales
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $(a,b)\in\mathbb{R}\times\mathbb{R}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $r>0$
+\end_inset
+
+, llamamos
+\begin_inset Formula 
+\begin{align*}
+J(a,b) & \coloneqq\begin{pmatrix}a & -b\\
+b & a
+\end{pmatrix},
+\end{align*}
+
+\end_inset
+
+con polinomio característico irreducible 
+\begin_inset Formula $p\coloneqq(X-a)^{2}+b^{2}$
+\end_inset
+
+, pues 
+\begin_inset Formula $p=X^{2}-2aX+a^{2}+b^{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $(-2a)^{2}-4(a^{2}+b^{2})=-b^{2}<0$
+\end_inset
+
+.
+ Entonces, para 
+\begin_inset Formula $r\in\mathbb{N}^{*}$
+\end_inset
+
+, llamamos 
+\series bold
+bloque de Jordan real
+\series default
+ de tamaño 
+\begin_inset Formula $r$
+\end_inset
+
+ asociado a 
+\begin_inset Formula $(a,b)$
+\end_inset
+
+ o a 
+\begin_inset Formula $p$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+J_{r}(a,b)\coloneqq\begin{pmatrix}\boxed{J(a,b)} & \boxed{I_{2}}\\
+ & \ddots & \ddots\\
+ &  & \ddots & \boxed{I_{2}}\\
+ &  &  & \boxed{J(a,b)}
+\end{pmatrix}\in{\cal M}_{2r}(\mathbb{R}).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Toda 
+\begin_inset Formula $C\in{\cal M}_{n}(\mathbb{R})$
+\end_inset
+
+ es semejante a una matriz de la forma
+\begin_inset Formula 
+\[
+\begin{pmatrix}\boxed{J_{r_{1}}(a_{1},b_{1})}\\
+ & \ddots\\
+ &  & \boxed{J_{r_{t}}(a_{t},b_{t})}\\
+ &  &  & \boxed{J_{h_{1}}(\lambda_{1})}\\
+ &  &  &  & \ddots\\
+ &  &  &  &  & \boxed{J_{h_{s}}(\lambda_{s})}
+\end{pmatrix},
+\]
+
+\end_inset
+
+única salvo reordenación de bloques, formada por
+\begin_inset Formula 
+\[
+\text{rk}((C-\lambda I)^{h-1})+\text{rk}((C-\lambda I)^{h+1})-2\text{rk}((C-\lambda I)^{h})
+\]
+
+\end_inset
+
+bloques 
+\begin_inset Formula $J_{h}(\lambda)$
+\end_inset
+
+ para cada 
+\begin_inset Formula $h\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ valor propio real de 
+\begin_inset Formula $C$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\frac{1}{2}(\text{rk}(p(C)^{r-1})+\text{rk}(p(C)^{r+1})-2\text{rk}(p(C)^{r})
+\]
+
+\end_inset
+
+bloques 
+\begin_inset Formula $J_{r}(a,b)$
+\end_inset
+
+ para cada 
+\begin_inset Formula $r\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $p=(X-a)^{2}+b^{2}$
+\end_inset
+
+ divisor irreducible cuadrático del polinomio característico de 
+\begin_inset Formula $C$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Por el teorema de clasificación de matrices cuadradas y el hecho de que
+ todos los irreducibles en 
+\begin_inset Formula $\mathbb{R}[X]$
+\end_inset
+
+ son de grado 1 o 2, solo hay que ver que 
+\begin_inset Formula $J_{r}(a,b)$
+\end_inset
+
+ es semejante a 
+\begin_inset Formula $C_{r}(p)$
+\end_inset
+
+, ambas con polinomio característico 
+\begin_inset Formula $p^{r}$
+\end_inset
+
+.
+ Pero si 
+\begin_inset Formula $J\coloneqq J_{r}(a,b)$
+\end_inset
+
+, 
+\begin_inset Formula $(J-aI)=J_{r}(0,b)$
+\end_inset
+
+ y, viendo 
+\begin_inset Formula $J_{r}(0,b)\in{\cal M}_{r}({\cal M}_{2}(K))$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+J_{r}(0,b)_{ij}=\begin{cases}
+J(0,b), & j=i;\\
+I_{2}, & j=i+1;\\
+0, & \text{en otro caso},
+\end{cases}
+\]
+
+\end_inset
+
+y como además 
+\begin_inset Formula $J(0,b)^{2}=-b^{2}I_{2}\in\text{GL}_{2}(\mathbb{R})$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+(J_{r}(0,b)^{2})_{ij}=\begin{cases}
+J(0,b)^{2}=-b^{2}I_{2}, & j=i;\\
+2J(0,b), & j=i+1;\\
+I_{2}, & j=i+2;\\
+0, & \text{en otro caso},
+\end{cases}
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $p(J)=(J-aI)^{2}+b^{2}$
+\end_inset
+
+ tiene la forma de la matriz del resultado anterior y 
+\begin_inset Formula $p(J)^{r}=0\neq p(J)^{r-1}$
+\end_inset
+
+.
+ Entonces el 
+\begin_inset Formula $\mathbb{R}[X]$
+\end_inset
+
+-módulo 
+\begin_inset Formula $M$
+\end_inset
+
+ asociado a 
+\begin_inset Formula $(\mathbb{R}^{2r},v\mapsto Jv)$
+\end_inset
+
+ tiene un sumando directo isomorfo a 
+\begin_inset Formula $\frac{\mathbb{R}[X]}{(p^{r})}$
+\end_inset
+
+, y como 
+\begin_inset Formula $\dim_{\mathbb{R}}\frac{\mathbb{R}[X]}{(p^{r})}=2h=\dim_{\mathbb{R}}M$
+\end_inset
+
+, 
+\begin_inset Formula $M\cong\frac{\mathbb{R}[X]}{(p^{r})}$
+\end_inset
+
+.
+ Pero por el teorema de clasificación de endomorfismos, 
+\begin_inset Formula $v\mapsto Jv$
+\end_inset
+
+ se expresa como 
+\begin_inset Formula $C_{r}(p)$
+\end_inset
+
+ en alguna base de 
+\begin_inset Formula $\mathbb{R}^{2r}$
+\end_inset
+
+ y por tanto en alguna de 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Series de Taylor pero en álgebra y son un porro
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+En realidad el porro es todo lo de antes.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\lambda\in K$
+\end_inset
+
+, 
+\begin_inset Formula $r,k\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $J\coloneqq J_{r}(\lambda)$
+\end_inset
+
+, si 
+\begin_inset Formula $k<r$
+\end_inset
+
+, 
+\begin_inset Formula $(J-\lambda I_{r})^{k}$
+\end_inset
+
+ tiene a 1 las celdas de la diagonal 
+\begin_inset Formula $k$
+\end_inset
+
+-ésima por encima de la diagonal principal y a 0 el resto, y si 
+\begin_inset Formula $k\geq r$
+\end_inset
+
+, 
+\begin_inset Formula $(J-\lambda I_{r})^{k}=0$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Esto equivale a que, en cualquier caso, 
+\begin_inset Formula $((J-\lambda I_{r})^{k})_{ij}\equiv\delta_{i-j,k}$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $k=1$
+\end_inset
+
+ esto es claro, y para 
+\begin_inset Formula $k>1$
+\end_inset
+
+, 
+\begin_inset Formula $((J-\lambda I_{r})^{k})_{ij}=\sum_{l=1}^{r}\delta_{i-l,k-1}\delta_{l-j,1}=\delta_{i-j,k}$
+\end_inset
+
+, pues lo de dentro del sumatorio vale 1 si y sólo si 
+\begin_inset Formula $i-l=k-1$
+\end_inset
+
+ y 
+\begin_inset Formula $l-j=1$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $l=j+1$
+\end_inset
+
+ e 
+\begin_inset Formula $i=j+k$
+\end_inset
+
+, pero si 
+\begin_inset Formula $j+k\leq r$
+\end_inset
+
+, 
+\begin_inset Formula $l\leq r$
+\end_inset
+
+ está dentro de rango y hay exactamente un sumando en que se da esto, y
+ si 
+\begin_inset Formula $j+k>r$
+\end_inset
+
+, esto no se da en ningún sumando pero tampoco se da 
+\begin_inset Formula $i-j=k$
+\end_inset
+
+ porque entonces sería 
+\begin_inset Formula $i>r$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+ igual a 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $D\subseteq\mathbb{K}$
+\end_inset
+
+ abierto, 
+\begin_inset Formula $\psi:D\to\mathbb{K}$
+\end_inset
+
+ infinitamente derivable, 
+\begin_inset Formula $\lambda\in D$
+\end_inset
+
+ y 
+\begin_inset Formula $J\coloneqq J_{r}(\lambda)$
+\end_inset
+
+, llamamos 
+\series bold
+valor
+\series default
+ o 
+\series bold
+evaluación
+\series default
+ de 
+\begin_inset Formula $\psi$
+\end_inset
+
+ en 
+\begin_inset Formula $J$
+\end_inset
+
+ a 
+\begin_inset Formula $\psi(J)$
+\end_inset
+
+, que es un polinomio en 
+\begin_inset Formula $J$
+\end_inset
+
+.
+ En efecto, 
+\begin_inset Formula $\psi$
+\end_inset
+
+ tiene una serie de Taylor 
+\begin_inset Formula $\psi(x)=\sum_{n\geq0}\frac{\psi^{(n)}(\lambda)}{n!}(x-\lambda)^{n}$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\psi(J)=\sum_{n\geq0}\frac{\psi^{(n)}(\lambda)}{n!}(J-\lambda I)^{n}$
+\end_inset
+
+, pero para 
+\begin_inset Formula $n\geq r$
+\end_inset
+
+ es 
+\begin_inset Formula $(J-\lambda I)^{n}=0$
+\end_inset
+
+, por lo que queda una suma finita que es un polinomio en 
+\begin_inset Formula $J$
+\end_inset
+
+.
+ Además:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $k\in\{1,\dots,r-1\}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+(J^{k})_{ij}=\binom{k}{j-i}\lambda^{k-j+i},
+\]
+
+\end_inset
+
+tomando el criterio 
+\begin_inset Formula $0\cdot\infty=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $k=1$
+\end_inset
+
+ es claro, pues para 
+\begin_inset Formula $j=i$
+\end_inset
+
+ es 
+\begin_inset Formula $J_{ij}=\lambda=\binom{1}{0}\lambda^{1}$
+\end_inset
+
+, para 
+\begin_inset Formula $j=i+1$
+\end_inset
+
+ es 
+\begin_inset Formula $J_{ij}=1=\binom{1}{1}\lambda^{0}$
+\end_inset
+
+ y en otro caso la fórmula da 0, usando el criterio si fuese necesario.
+ Para 
+\begin_inset Formula $k>1$
+\end_inset
+
+, por inducción,
+\begin_inset Formula 
+\begin{align*}
+(J^{k})_{ij} & =\sum_{l=1}^{r}(J^{k-1})_{il}J_{lj}=\sum_{l=1}^{r}\binom{k-1}{l-i}\binom{1}{j-l}\lambda^{(k-1-l+i)+(1-j+l)}=\\
+ & =\sum_{l}\binom{k-1}{l-i}\binom{1}{(j-i)-(l-i)}\lambda^{k+i-j}=\binom{k}{j-i}\lambda^{k+i-j},
+\end{align*}
+
+\end_inset
+
+donde justificamos expandir el rango del sumatorio viendo que, si 
+\begin_inset Formula $0\leq l-i\leq k-1$
+\end_inset
+
+ y 
+\begin_inset Formula $0\leq j-l\leq1$
+\end_inset
+
+, entonces por lo primero 
+\begin_inset Formula $i\leq l$
+\end_inset
+
+ y por lo segundo 
+\begin_inset Formula $l\leq j$
+\end_inset
+
+, luego 
+\begin_inset Formula $l\in\{1,\dots,r\}$
+\end_inset
+
+.
+ 
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+(\psi(J))_{ij}=\begin{cases}
+\frac{\psi^{(j-i)}(\lambda)}{(j-i)!}, & j\geq i;\\
+0, & \text{en otro caso}.
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\psi(J)=\sum_{n\geq0}\frac{\psi^{(n)}(\lambda)}{n!}(J-\lambda I)^{n}$
+\end_inset
+
+, con lo que
+\begin_inset Formula 
+\[
+(\psi(J))_{ij}=\sum_{n\geq0}\frac{\psi^{(n)}(\lambda)}{n!}\delta_{j-i,n}=\begin{cases}
+\frac{\psi^{(n)}(\lambda)}{n!}, & n\coloneqq j-i\geq0;\\
+0, & \text{en otro caso}.
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $C\in{\cal M}_{n}(\mathbb{K})$
+\end_inset
+
+ y 
+\begin_inset Formula $P\in\text{GL}_{n}(\mathbb{K})$
+\end_inset
+
+ son tales que 
+\begin_inset Formula $P^{-1}CP\eqqcolon\text{diag}(J_{1},\dots,J_{t})$
+\end_inset
+
+ con los 
+\begin_inset Formula $J_{i}$
+\end_inset
+
+ bloques de Jordan, 
+\begin_inset Formula $D\subseteq\mathbb{K}$
+\end_inset
+
+ es un abierto que contiene a todos los valores propios de 
+\begin_inset Formula $C$
+\end_inset
+
+ y 
+\begin_inset Formula $\psi:D\to\mathbb{K}$
+\end_inset
+
+ es infinitamente derivable, llamamos 
+\series bold
+valor
+\series default
+ o 
+\series bold
+evaluación
+\series default
+ de 
+\begin_inset Formula $\psi$
+\end_inset
+
+ en 
+\begin_inset Formula $C$
+\end_inset
+
+ a 
+\begin_inset Formula $\psi(C)\coloneqq P(\psi(J_{1})\oplus\dots\oplus\psi(J_{t}))P^{-1}$
+\end_inset
+
+, que no depende de la 
+\begin_inset Formula $P$
+\end_inset
+
+ elegida.
+\end_layout
+
+\end_body
+\end_document
diff --git a/ac/na.lyx b/ac/na.lyx
new file mode 100644
index 0000000..1f36678
--- /dev/null
+++ b/ac/na.lyx
@@ -0,0 +1,1250 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+orden
+\series default
+ de [un grupo] 
+\begin_inset Formula $G$
+\end_inset
+
+ al cardinal del conjunto.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un anillo, 
+\begin_inset Formula $(A,+)$
+\end_inset
+
+ es su 
+\series bold
+grupo aditivo
+\series default
+, que es abeliano, y 
+\begin_inset Formula $(A^{*},\cdot)$
+\end_inset
+
+ es su 
+\series bold
+grupo de unidades
+\series default
+, que es abeliano cuando el anillo es conmutativo.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+orden
+\series default
+ de 
+\begin_inset Formula $a\in G$
+\end_inset
+
+ al orden de 
+\begin_inset Formula $\langle a\rangle$
+\end_inset
+
+, 
+\begin_inset Formula $|a|\coloneqq|\langle a\rangle|$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $\langle a\rangle_{n}$
+\end_inset
+
+ para referirnos a 
+\begin_inset Formula $\langle a\rangle$
+\end_inset
+
+ indicando que tiene orden 
+\begin_inset Formula $n$
+\end_inset
+
+.
+ El orden de 
+\begin_inset Formula $a$
+\end_inset
+
+ divide al de 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $f:\mathbb{Z}\to G$
+\end_inset
+
+ el homomorfismo dado por 
+\begin_inset Formula $f(n)\coloneqq a^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $\ker f=n\mathbb{Z}$
+\end_inset
+
+ para algún 
+\begin_inset Formula $n\geq0$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $n=0$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectivo y 
+\begin_inset Formula $(\mathbb{Z},+)\cong\langle a\rangle$
+\end_inset
+
+, y en otro caso 
+\begin_inset Formula $\mathbb{Z}_{n}\cong\langle a\rangle$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $n=|a|$
+\end_inset
+
+ y 
+\begin_inset Formula $a^{n}=1\iff|a|\mid n$
+\end_inset
+
+.
+ De aquí, 
+\begin_inset Formula $a^{k}=a^{l}\iff k\equiv l\bmod n$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $|a|$
+\end_inset
+
+ es el menor entero positivo con 
+\begin_inset Formula $a^{n}=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $a$
+\end_inset
+
+ tiene orden finito y 
+\begin_inset Formula $n>0$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+|a^{n}|=\frac{|a|}{\text{mcd}\{|a|,n\}}.
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $G=\langle a\rangle$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ tiene orden infinito, 
+\begin_inset Formula $G\cong(\mathbb{Z},+)\cong C_{\infty}$
+\end_inset
+
+ y los subgrupos de 
+\begin_inset Formula $G$
+\end_inset
+
+ son los 
+\begin_inset Formula $\langle a^{n}\rangle$
+\end_inset
+
+ con 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $|G|=n$
+\end_inset
+
+, 
+\begin_inset Formula $G\cong(\mathbb{Z}_{n},+)\cong C_{n}$
+\end_inset
+
+ y los subgrupos de 
+\begin_inset Formula $G$
+\end_inset
+
+ son exactamente uno de orden 
+\begin_inset Formula $d$
+\end_inset
+
+ por cada 
+\begin_inset Formula $d\mid n$
+\end_inset
+
+, 
+\begin_inset Formula $\langle a^{n/d}\rangle_{d}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Todos los subgrupos y grupos cociente de 
+\begin_inset Formula $G$
+\end_inset
+
+ son cíclicos.
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $p\in\mathbb{N}$
+\end_inset
+
+ es primo, todos los grupos de orden 
+\begin_inset Formula $p$
+\end_inset
+
+ son isomorfos a 
+\begin_inset Formula $(\mathbb{Z}_{p},+)$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $G=\langle g_{1},\dots,g_{n}\rangle$
+\end_inset
+
+ y 
+\begin_inset Formula $N\unlhd G$
+\end_inset
+
+, 
+\begin_inset Formula $G/N=\langle g_{1}N,\dots,g_{n}N\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema chino de los restos para grupos:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $H$
+\end_inset
+
+ son subgrupos cíclicos de órdenes respectivos 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+, 
+\begin_inset Formula $G\times H$
+\end_inset
+
+ es cíclico si y sólo si 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+ son coprimos.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $g,h\in G$
+\end_inset
+
+ tienen órdenes respectivos 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+ coprimos y 
+\begin_inset Formula $gh=hg$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\langle g,h\rangle$
+\end_inset
+
+ es cíclico de orden 
+\begin_inset Formula $nm$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dados un grupo 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, llamamos 
+\series bold
+conjugado
+\series default
+ de 
+\begin_inset Formula $g\in G$
+\end_inset
+
+ por 
+\begin_inset Formula $a$
+\end_inset
+
+ a 
+\begin_inset Formula $g^{a}\coloneqq a^{-1}ga$
+\end_inset
+
+, y conjugado de 
+\begin_inset Formula $X\subseteq G$
+\end_inset
+
+ por 
+\begin_inset Formula $a$
+\end_inset
+
+ a 
+\begin_inset Formula $X^{a}\coloneqq\{x^{a}\}_{x\in X}$
+\end_inset
+
+.
+ Dos elementos 
+\begin_inset Formula $x,y\in G$
+\end_inset
+
+ o conjuntos 
+\begin_inset Formula $x,y\subseteq G$
+\end_inset
+
+ son 
+\series bold
+conjugados
+\series default
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ si existe 
+\begin_inset Formula $a\in G$
+\end_inset
+
+ con 
+\begin_inset Formula $x^{a}=y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, llamamos 
+\series bold
+automorfismo interno
+\series default
+ definido por 
+\begin_inset Formula $a$
+\end_inset
+
+ al automorfismo 
+\begin_inset Formula $\iota_{a}:G\to G$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\iota_{a}(x)\coloneqq x^{a}$
+\end_inset
+
+.
+ Su inverso es 
+\begin_inset Formula $\iota_{a^{-1}}$
+\end_inset
+
+.
+ El conjugado por 
+\begin_inset Formula $a$
+\end_inset
+
+ de un subgrupo de 
+\begin_inset Formula $G$
+\end_inset
+
+ es otro subgrupo de 
+\begin_inset Formula $G$
+\end_inset
+
+ del mismo orden.
+ [...]
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall g,a,b\in G,g^{ab}=(g^{a})^{b}$
+\end_inset
+
+, y [...] la relación de ser conjugados es de equivalencia.
+ Las clases de equivalencia se llaman 
+\series bold
+clases de conjugación
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+, y llamamos 
+\begin_inset Formula $a^{G}\coloneqq[a]=\{a^{g}\}_{g\in G}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $X$
+\end_inset
+
+ un conjunto.
+ Una 
+\series bold
+acción por la izquierda
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ es una función 
+\begin_inset Formula $\cdot:G\times X\to X$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in X,(\forall g,h\in G,(gh)\cdot x=g\cdot(h\cdot x)\land1\cdot x=x)$
+\end_inset
+
+, y una 
+\series bold
+acción por la derecha
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ es una función 
+\begin_inset Formula $\cdot:X\times G\to X$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall x\in X,(\forall g,h\in G,x\cdot(gh)=(x\cdot g)\cdot h\land x\cdot1=x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\cdot:G\times X\to X$
+\end_inset
+
+ es una acción por la izquierda de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, llamamos 
+\series bold
+órbita
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $G\cdot x\coloneqq\{g\cdot x\}_{g\in G}$
+\end_inset
+
+ y 
+\series bold
+estabilizador
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{Estab}_{G}(x)\coloneqq\{g\in G\mid g\cdot x=x\}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\cdot:X\times G\to X$
+\end_inset
+
+ es una acción por la derecha de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $X$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, llamamos órbita de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $x\cdot G\coloneqq\{x\cdot g\}_{g\in G}$
+\end_inset
+
+ y estabilizador de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{Estab}_{G}(x)\coloneqq\{g\in G\mid x\cdot g=x\}$
+\end_inset
+
+.
+ Las órbitas forman una partición de 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Llamamos 
+\series bold
+acción por traslación a la izquierda
+\series default
+ a la acción por la izquierda de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $G/H$
+\end_inset
+
+ dada por 
+\begin_inset Formula $g\cdot xH=gxH$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $G\cdot xH=G/H$
+\end_inset
+
+ y 
+\begin_inset Formula 
+\[
+\text{Estab}_{G}(xH)=[...]=H^{x^{-1}}.
+\]
+
+\end_inset
+
+Análogamente llamamos 
+\series bold
+acción por traslación a la derecha
+\series default
+ a la acción por la derecha de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $H\backslash G$
+\end_inset
+
+ dada por 
+\begin_inset Formula $Hx\cdot g=Hxg$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Cuando 
+\begin_inset Formula $H=1$
+\end_inset
+
+, la acción de traslación es de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+, con 
+\begin_inset Formula $G\cdot x=G$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Estab}_{G}(x)=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La 
+\series bold
+acción por conjugación
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $G$
+\end_inset
+
+ es la acción por la derecha 
+\begin_inset Formula $x\cdot g\coloneqq x^{g}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $x\cdot G=x^{G}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Estab}_{G}(x)=C_{G}(x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $S$
+\end_inset
+
+ es el conjunto de subgrupos de 
+\begin_inset Formula $G$
+\end_inset
+
+, la 
+\series bold
+acción por conjugación de 
+\begin_inset Formula $G$
+\end_inset
+
+ en sus subgrupos
+\series default
+ es la acción por la derecha de 
+\begin_inset Formula $G$
+\end_inset
+
+ en 
+\begin_inset Formula $S$
+\end_inset
+
+ 
+\begin_inset Formula $H\cdot g=H^{g}$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $X$
+\end_inset
+
+ es un conjunto, 
+\begin_inset Formula $\cdot:S_{n}\times X^{n}\to X^{n}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\sigma\cdot(x_{1},\dots,x_{n})\coloneqq(x_{\sigma(1)},\dots,x_{\sigma(n)})$
+\end_inset
+
+ es una acción por la izquierda.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $\cdot:G\times X\to X$
+\end_inset
+
+ una acción por la izquierda, 
+\begin_inset Formula $H\leq G$
+\end_inset
+
+ e 
+\begin_inset Formula $Y\subseteq X$
+\end_inset
+
+, si 
+\begin_inset Formula $\forall h\in H,y\in Y,h\cdot y\in Y$
+\end_inset
+
+, 
+\begin_inset Formula $\cdot|_{H\times Y}$
+\end_inset
+
+ es una acción por la izquierda de 
+\begin_inset Formula $H$
+\end_inset
+
+ en 
+\begin_inset Formula $Y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $G$
+\end_inset
+
+ un grupo actuando sobre un conjunto 
+\begin_inset Formula $X$
+\end_inset
+
+, 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ y 
+\begin_inset Formula $g\in G$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{Estab}_{G}(x)\leq G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $[G:\text{Estab}_{G}(x)]=|G\cdot x|$
+\end_inset
+
+.
+ En particular, si 
+\begin_inset Formula $G$
+\end_inset
+
+ es finito, 
+\begin_inset Formula $|G\cdot x|\mid|G|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si la acción es por la izquierda, 
+\begin_inset Formula $\text{Estab}_{G}(g\cdot x)=\text{Estab}_{G}(x)^{g^{-1}}$
+\end_inset
+
+, y si es por la derecha, 
+\begin_inset Formula $\text{Estab}_{G}(x\cdot g)=\text{Estab}_{G}(x)^{g}$
+\end_inset
+
+.
+ En particular, si 
+\begin_inset Formula $x,g\in G$
+\end_inset
+
+ y 
+\begin_inset Formula $H\leq G$
+\end_inset
+
+, 
+\begin_inset Formula $C_{G}(x^{g})=C_{G}(x)^{g}$
+\end_inset
+
+ y 
+\begin_inset Formula $N_{G}(H^{g})=N_{G}(H)^{g}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $R$
+\end_inset
+
+ es un conjunto irredundante de representantes de las órbitas, 
+\begin_inset Formula $|X|=\sum_{r\in R}|G\cdot r|=\sum_{r\in R}[G:\text{Estab}_{G}(r)]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $G$
+\end_inset
+
+ es un grupo y 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, 
+\begin_inset Formula $|a^{G}|=[G:C_{G}(a)]$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $a^{G}$
+\end_inset
+
+ es unipuntual si y sólo si 
+\begin_inset Formula $a\in Z(G)$
+\end_inset
+
+.
+ 
+\series bold
+Ecuación de clases:
+\series default
+ Si 
+\begin_inset Formula $G$
+\end_inset
+
+ es finito y 
+\begin_inset Formula $X\subseteq G$
+\end_inset
+
+ contiene exactamente un elemento de cada clase de conjugación con al menos
+ dos elementos, entonces 
+\begin_inset Formula $|G|=|Z(G)|+\sum_{x\in X}[G:C_{G}(x)]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado un número primo 
+\begin_inset Formula $p$
+\end_inset
+
+, un 
+\series bold
+
+\begin_inset Formula $p$
+\end_inset
+
+-grupo
+\series default
+ es un grupo en que todo elemento tiene orden potencia de 
+\begin_inset Formula $p$
+\end_inset
+
+, y un grupo finito es un 
+\begin_inset Formula $p$
+\end_inset
+
+-grupo si y sólo si su orden es potencia de 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema de Cauchy:
+\series default
+ Si 
+\begin_inset Formula $G$
+\end_inset
+
+ es un grupo finito con orden múltiplo de un primo 
+\begin_inset Formula $p$
+\end_inset
+
+, 
+\begin_inset Formula $G$
+\end_inset
+
+ tiene un elemento de orden 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dados un grupo finito 
+\begin_inset Formula $G$
+\end_inset
+
+ y un número primo 
+\begin_inset Formula $p$
+\end_inset
+
+, 
+\begin_inset Formula $H\leq G$
+\end_inset
+
+ es un 
+\series bold
+
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de Sylow
+\series default
+ de 
+\begin_inset Formula $G$
+\end_inset
+
+ si es un 
+\begin_inset Formula $p$
+\end_inset
+
+-grupo y 
+\begin_inset Formula $[G:H]$
+\end_inset
+
+ es coprimo con 
+\begin_inset Formula $p$
+\end_inset
+
+, si y sólo si es un 
+\begin_inset Formula $p$
+\end_inset
+
+-grupo y 
+\begin_inset Formula $|H|$
+\end_inset
+
+ es la mayor potencia de 
+\begin_inset Formula $p$
+\end_inset
+
+ que divide a 
+\begin_inset Formula $|G|$
+\end_inset
+
+.
+ Llamamos 
+\begin_inset Formula $s_{p}(G)$
+\end_inset
+
+ al número de 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupos de Sylow de 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teoremas de Sylow:
+\series default
+ Sean 
+\begin_inset Formula $p$
+\end_inset
+
+ un número primo y 
+\begin_inset Formula $G$
+\end_inset
+
+ un grupo finito de orden 
+\begin_inset Formula $n\coloneqq p^{k}m$
+\end_inset
+
+ para ciertos 
+\begin_inset Formula $k,m\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $p\nmid m$
+\end_inset
+
+.
+ Entonces:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $G$
+\end_inset
+
+ tiene al menos un 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de Sylow, que tendrá orden 
+\begin_inset Formula $p^{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $P$
+\end_inset
+
+ es un 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de Sylow de 
+\begin_inset Formula $G$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ es un 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupo de 
+\begin_inset Formula $G$
+\end_inset
+
+, existe 
+\begin_inset Formula $g\in G$
+\end_inset
+
+ tal que 
+\begin_inset Formula $Q\subseteq P^{g}$
+\end_inset
+
+.
+ En particular, todos los 
+\begin_inset Formula $p$
+\end_inset
+
+-subgrupos de Sylow de 
+\begin_inset Formula $G$
+\end_inset
+
+ son conjugados en 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $s_{p}(G)\mid m$
+\end_inset
+
+ y 
+\begin_inset Formula $s_{p}(G)\equiv1\bmod p$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/ac/nb.lyx b/ac/nb.lyx
new file mode 100644
index 0000000..0db7de6
--- /dev/null
+++ b/ac/nb.lyx
@@ -0,0 +1,2735 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Cuerpos de fracciones
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $D\neq0$
+\end_inset
+
+ un dominio y 
+\begin_inset Formula $X\coloneqq D\times(D\setminus\{0\})$
+\end_inset
+
+, definimos la relación binaria 
+\begin_inset Formula 
+\[
+(a_{1},s_{1})\sim(a_{2},s_{2}):\iff a_{1}s_{2}=a_{2}s_{1}.
+\]
+
+\end_inset
+
+ Esta relación es de equivalencia.
+ Llamamos 
+\begin_inset Formula $a/s\coloneqq\frac{a}{s}\coloneqq[(a,s)]\in Q(D)\coloneqq X/\sim$
+\end_inset
+
+, y las operaciones
+\begin_inset Formula 
+\begin{align*}
+\frac{a_{1}}{s_{1}}+\frac{a_{2}}{s_{2}} & :=\frac{a_{1}s_{2}+a_{2}s_{1}}{s_{1}s_{2}}, & \frac{a_{1}}{s_{1}}\cdot\frac{a_{2}}{s_{2}} & :=\frac{a_{1}a_{2}}{s_{1}s_{2}},
+\end{align*}
+
+\end_inset
+
+están bien definidas.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $a,b\in D$
+\end_inset
+
+ y 
+\begin_inset Formula $s,t\in D\setminus\{0\}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{a}{s}=\frac{0}{1}\iff a=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{a}{s}=\frac{1}{1}\iff a=s$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{at}{st}=\frac{a}{s}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{a}{s}=\frac{b}{s}\iff a=b$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{a}{s}+\frac{b}{s}=\frac{a+b}{s}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] 
+\begin_inset Formula $(Q(D),+,\cdot)$
+\end_inset
+
+ es un cuerpo llamado 
+\series bold
+cuerpo de fracciones
+\series default
+ o 
+\series bold
+de cocientes
+\series default
+ de 
+\begin_inset Formula $D$
+\end_inset
+
+ cuyo cero es 
+\begin_inset Formula $\frac{0}{1}$
+\end_inset
+
+ y cuyo uno es 
+\begin_inset Formula $\frac{1}{1}$
+\end_inset
+
+ .
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es el cuerpo de fracciones de 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+.
+ [...] 
+\begin_inset Formula $u:D\to Q(D)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $u(a)\coloneqq a/1$
+\end_inset
+
+ es un homomorfismo inyectivo, por lo que podemos ver a 
+\begin_inset Formula $D$
+\end_inset
+
+ como un subdominio de 
+\begin_inset Formula $Q(D)$
+\end_inset
+
+ identificando a cada 
+\begin_inset Formula $a\in D$
+\end_inset
+
+ con 
+\begin_inset Formula $a/1\in Q(D)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Propiedad universal del cuerpo de fracciones:
+\series default
+ Dados un dominio 
+\begin_inset Formula $D$
+\end_inset
+
+ y 
+\begin_inset Formula $u:D\to Q(D)$
+\end_inset
+
+ dada por 
+\begin_inset Formula $u(a)\coloneqq a/1$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo y 
+\begin_inset Formula $f:D\to K$
+\end_inset
+
+ un homomorfismo inyectivo, el único homomorfismo de cuerpos 
+\begin_inset Formula $\tilde{f}:Q(D)\to K$
+\end_inset
+
+ con 
+\begin_inset Formula $\tilde{f}\circ u=f$
+\end_inset
+
+ viene dado por 
+\begin_inset Formula $\tilde{f}(\frac{a}{s})=f(a)f(s)^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo no trivial y 
+\begin_inset Formula $g,h:Q(D)\to K$
+\end_inset
+
+ homomorfismos que coinciden en 
+\begin_inset Formula $D$
+\end_inset
+
+, entonces 
+\begin_inset Formula $g=h$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $F$
+\end_inset
+
+ un cuerpo no trivial y 
+\begin_inset Formula $v:D\to F$
+\end_inset
+
+ un homomorfismo inyectivo tal que para todo cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+ y homomorfismo inyectivo 
+\begin_inset Formula $f:D\to K$
+\end_inset
+
+ existe un único homomorfismo 
+\begin_inset Formula $\tilde{f}:F\to K$
+\end_inset
+
+ con 
+\begin_inset Formula $\tilde{f}\circ v=f$
+\end_inset
+
+, entonces existe un isomorfismo 
+\begin_inset Formula $\phi:F\to Q(D)$
+\end_inset
+
+ con 
+\begin_inset Formula $\phi\circ v=u$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $D$
+\end_inset
+
+ un dominio, 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo no trivial y 
+\begin_inset Formula $f:D\to K$
+\end_inset
+
+ un homomorfismo inyectivo, 
+\begin_inset Formula $K$
+\end_inset
+
+ contiene un subcuerpo isomorfo a 
+\begin_inset Formula $Q(D)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+De aquí, para 
+\begin_inset Formula $m\in\mathbb{Z}$
+\end_inset
+
+, 
+\begin_inset Formula $Q(\mathbb{Z}[\sqrt{m}])\cong\mathbb{Q}[\sqrt{m}]$
+\end_inset
+
+, lo que nos permite identificar los elementos de 
+\begin_inset Formula $Q(\mathbb{Z}[\sqrt{m}])$
+\end_inset
+
+ con los de 
+\begin_inset Formula $\mathbb{Q}[\sqrt{m}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo no trivial, existe un subcuerpo 
+\begin_inset Formula $K'$
+\end_inset
+
+ de 
+\begin_inset Formula $K$
+\end_inset
+
+ llamado 
+\series bold
+subcuerpo primo
+\series default
+ de 
+\begin_inset Formula $K$
+\end_inset
+
+ contenido en cualquier subcuerpo de 
+\begin_inset Formula $K$
+\end_inset
+
+, y este es isomorfo a 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ si la característica de 
+\begin_inset Formula $K$
+\end_inset
+
+ es un entero primo 
+\begin_inset Formula $p$
+\end_inset
+
+ o a 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ en caso contrario.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Polinomios
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ es un subanillo de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ identificando los elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ con los 
+\series bold
+polinomios constantes
+\series default
+, de la forma 
+\begin_inset Formula $P(X)=a_{0}$
+\end_inset
+
+.
+ Dado un ideal 
+\begin_inset Formula $I$
+\end_inset
+
+ de 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0}\in I\}$
+\end_inset
+
+ e 
+\begin_inset Formula $I[X]\coloneqq\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0},\dots,a_{n}\in I\}$
+\end_inset
+
+ son ideales de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $p\coloneqq\sum_{k\in\mathbb{N}}p_{k}X^{k}\in A[X]\setminus\{0\}$
+\end_inset
+
+, llamamos 
+\series bold
+grado
+\series default
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{gr}(p)\coloneqq\max\{k\in\mathbb{N}\mid p_{k}\neq0\}$
+\end_inset
+
+, 
+\series bold
+coeficiente
+\series default
+ de 
+\series bold
+grado
+\series default
+ 
+\begin_inset Formula $k$
+\end_inset
+
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+ a 
+\begin_inset Formula $p_{k}$
+\end_inset
+
+, 
+\series bold
+coeficiente independiente
+\series default
+ al de grado 0 y 
+\series bold
+coeficiente principal
+\series default
+ al de grado 
+\begin_inset Formula $\text{gr}(p)$
+\end_inset
+
+.
+ Un polinomio es 
+\series bold
+mónico
+\series default
+ si su coeficiente principal es 1.
+ El polinomio 0 tiene grado 
+\begin_inset Formula $-\infty$
+\end_inset
+
+ por convención.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+monomio
+\series default
+ es un polinomio de la forma 
+\begin_inset Formula $aX^{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+.
+ Todo polinomio en 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ se escribe como suma finita de monomios de distinto grado de forma única
+ salvo orden.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $P,Q\in A[X]\setminus\{0\}$
+\end_inset
+
+ tienen coeficientes principales respectivos 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $q$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{gr}(P+Q)\leq\max\{\text{gr}(P),\text{gr}(Q)\}$
+\end_inset
+
+, con desigualdad estricta si y sólo si 
+\begin_inset Formula $\text{gr}(P)=\text{gr}(Q)$
+\end_inset
+
+ y 
+\begin_inset Formula $p+q=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\text{gr}(PQ)\leq\text{gr}(P)+\text{gr}(Q)$
+\end_inset
+
+, con igualdad si y sólo si 
+\begin_inset Formula $pq\neq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A[X]$
+\end_inset
+
+ no es un cuerpo.
+ Es un dominio si y sólo si lo es 
+\begin_inset Formula $A$
+\end_inset
+
+, en cuyo caso llamamos 
+\series bold
+cuerpo de las funciones racionales
+\series default
+ sobre 
+\begin_inset Formula $A$
+\end_inset
+
+ al cuerpo de fracciones de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] 
+\series bold
+Propiedad universal del anillo de polinomios
+\series default
+ (
+\series bold
+PUAP
+\series default
+)
+\series bold
+:
+\series default
+ Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo y 
+\begin_inset Formula $u:A\to A[X]$
+\end_inset
+
+ el homomorfismo inclusión:
+\end_layout
+
+\begin_layout Enumerate
+Para cada homomorfismo de anillos conmutativos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, el único homomorfismo 
+\begin_inset Formula $\tilde{f}:A[X]\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}(X)=b$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}\circ u=f$
+\end_inset
+
+ es
+\begin_inset Formula 
+\[
+\tilde{f}\left(\sum_{n}p_{n}X^{n}\right):=\sum_{n}f(p_{n})b^{n}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $u$
+\end_inset
+
+ están determinados salvo isomorfismos por la propiedad universal: dados
+ un homomorfismo de anillos 
+\begin_inset Formula $v:A\to P$
+\end_inset
+
+ y 
+\begin_inset Formula $t\in P$
+\end_inset
+
+ tales que, para cada homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, existe un único 
+\begin_inset Formula $\tilde{f}:P\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}\circ v=f$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}(t)=b$
+\end_inset
+
+, existe un isomorfismo 
+\begin_inset Formula $\phi:A[X]\to P$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\phi\circ u=v$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi(X)=t$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+, el 
+\series bold
+homomorfismo de sustitución
+\series default
+ o 
+\series bold
+de evaluación
+\series default
+ en 
+\begin_inset Formula $b$
+\end_inset
+
+ es 
+\begin_inset Formula $S_{b}:A[X]\to B$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+S_{b}(p):=p(b):=\sum_{n}p_{n}b^{n},
+\]
+
+\end_inset
+
+y su imagen es el subanillo generado por 
+\begin_inset Formula $A\cup\{b\}$
+\end_inset
+
+, llamado 
+\begin_inset Formula $A[b]$
+\end_inset
+
+.
+ Todo 
+\begin_inset Formula $p\in A[X]$
+\end_inset
+
+ induce una 
+\series bold
+función polinómica
+\series default
+ 
+\begin_inset Formula $\hat{p}:B\to B$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\hat{p}(b)\coloneqq S_{b}(p)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dado 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, el homomorfismo de sustitución 
+\begin_inset Formula $S_{X+a}$
+\end_inset
+
+ es un automorfismo de 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ con inverso 
+\begin_inset Formula $S_{X-a}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un anillo conmutativo, 
+\begin_inset Formula $\frac{A[X]}{(X)}\cong A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Todo homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ induce un homomorfismo 
+\begin_inset Formula $\hat{f}:A[X]\to B[X]$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+\hat{f}(p)=\sum_{n}f(p_{n})X^{n},
+\]
+
+\end_inset
+
+que es inyectivo o suprayectivo si lo es 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+, 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ lo es de 
+\begin_inset Formula $B[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $I$
+\end_inset
+
+ es un ideal de 
+\begin_inset Formula $A$
+\end_inset
+
+, el 
+\series bold
+homomorfismo de reducción de coeficientes módulo 
+\begin_inset Formula $I$
+\end_inset
+
+
+\series default
+ es 
+\begin_inset Formula $\tilde{\pi}:A[X]\to(A/I)[X]$
+\end_inset
+
+ dado por
+\begin_inset Formula 
+\[
+\tilde{\pi}(p):=\sum_{n}(p_{n}+I)X^{n}.
+\]
+
+\end_inset
+
+Su núcleo es 
+\begin_inset Formula $I[X]$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $(A/I)[X]\cong\frac{A[X]}{I[X]}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage pagebreak
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Descomposiciones de polinomios en dominios
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f,g\in A[X]$
+\end_inset
+
+, si el coeficiente principal de 
+\begin_inset Formula $g$
+\end_inset
+
+ es invertible en 
+\begin_inset Formula $A$
+\end_inset
+
+, existen dos únicos polinomios 
+\begin_inset Formula $q,r\in A[X]$
+\end_inset
+
+, llamados respectivamente 
+\series bold
+cociente
+\series default
+ y 
+\series bold
+resto
+\series default
+ de la 
+\series bold
+división
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ entre 
+\begin_inset Formula $g$
+\end_inset
+
+, tales que 
+\begin_inset Formula $f=gq+r$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(r)<\text{gr}(g)$
+\end_inset
+
+ [...].
+ En particular, el grado es una función euclídea.
+ 
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema del resto:
+\series default
+ Dados 
+\begin_inset Formula $f\in A[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, el resto de 
+\begin_inset Formula $f$
+\end_inset
+
+ entre 
+\begin_inset Formula $X-a$
+\end_inset
+
+ es 
+\begin_inset Formula $f(a)$
+\end_inset
+
+.
+ De aquí se obtiene el 
+\series bold
+teorema de Ruffini
+\series default
+, que dice que 
+\begin_inset Formula $f$
+\end_inset
+
+ es divisible por 
+\begin_inset Formula $X-a$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $f(a)=0$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $a$
+\end_inset
+
+ es una 
+\series bold
+raíz
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $f\in A[X]\setminus\{0\}$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, existe 
+\begin_inset Formula $m\coloneqq\max\{k\in\mathbb{N}\mid(X-a)^{k}\mid f\}$
+\end_inset
+
+.
+ Llamamos a 
+\begin_inset Formula $m$
+\end_inset
+
+ 
+\series bold
+multiplicidad
+\series default
+ de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $f$
+\end_inset
+
+, y 
+\begin_inset Formula $a$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $f$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $m\geq1$
+\end_inset
+
+.
+ Decimos que 
+\begin_inset Formula $a$
+\end_inset
+
+ es una 
+\series bold
+raíz simple
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ si 
+\begin_inset Formula $m=1$
+\end_inset
+
+ y que es una 
+\series bold
+raíz compuesta
+\series default
+ si 
+\begin_inset Formula $m>1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La multiplicidad de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $f$
+\end_inset
+
+ es el único natural 
+\begin_inset Formula $m$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f=(X-a)^{m}g$
+\end_inset
+
+ para algún 
+\begin_inset Formula $g\in A[X]$
+\end_inset
+
+ del que 
+\begin_inset Formula $a$
+\end_inset
+
+ no es raíz.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un dominio, 
+\begin_inset Formula $f\in D[X]\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $a_{1},\dots,a_{n}$
+\end_inset
+
+ son 
+\begin_inset Formula $n$
+\end_inset
+
+ elementos de 
+\begin_inset Formula $D$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}\in\mathbb{Z}^{>0}$
+\end_inset
+
+ con 
+\begin_inset Formula $(X-a_{k})^{\alpha_{k}}\mid f$
+\end_inset
+
+ para cada 
+\begin_inset Formula $k$
+\end_inset
+
+, entonces 
+\begin_inset Formula $(X-a_{1})^{\alpha_{1}}\cdots(X-a_{n})^{\alpha_{n}}\mid f$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\sum_{k=1}^{n}\alpha_{k}\leq\text{gr}(f)$
+\end_inset
+
+ y, en particular, la suma de las multiplicidades de las raíces de 
+\begin_inset Formula $f$
+\end_inset
+
+, y el número de raíces, no son superiores a 
+\begin_inset Formula $\text{gr}(f)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Principio de las identidades polinómicas:
+\series default
+ Sea 
+\begin_inset Formula $D$
+\end_inset
+
+ un dominio:
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $f,g\in D[X]$
+\end_inset
+
+, si las funciones polinómicas 
+\begin_inset Formula $f,g:D\to D$
+\end_inset
+
+ coinciden en 
+\begin_inset Formula $m$
+\end_inset
+
+ elementos de 
+\begin_inset Formula $D$
+\end_inset
+
+ con 
+\begin_inset Formula $m>\text{gr}(f),\text{gr}(g)$
+\end_inset
+
+, los polinomios 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ son iguales.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $D$
+\end_inset
+
+ es infinito si y sólo si cualquier par de polinomios distintos en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ define dos funciones polinómicas distintas en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como ejemplo de lo anterior, por el teorema pequeño de Fermat, dado un primo
+ 
+\begin_inset Formula $p$
+\end_inset
+
+, todos los elementos de 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ son raíces de 0 y 
+\begin_inset Formula $X^{p}-X$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado un anillo conmutativo 
+\begin_inset Formula $A$
+\end_inset
+
+, definimos la 
+\series bold
+derivada
+\series default
+ de 
+\begin_inset Formula $P\coloneqq\sum_{k}a_{k}X^{k}\in A[X]$
+\end_inset
+
+ como 
+\begin_inset Formula $P'\coloneqq D(P)\coloneqq\sum_{k\geq1}ka_{k}X^{k-1}$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $P^{(0)}\coloneqq P$
+\end_inset
+
+ y 
+\begin_inset Formula $P^{(n+1)}\coloneqq P^{(n)\prime}$
+\end_inset
+
+.
+ Dados 
+\begin_inset Formula $a,b\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $P,Q\in A[X]$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(aP+bQ)'=aP'+bQ'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(PQ)'=P'Q+PQ'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(P^{n})'=nP^{n-1}P'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados un dominio 
+\begin_inset Formula $D$
+\end_inset
+
+ de característica 0, 
+\begin_inset Formula $P\in D[X]\setminus\{0\}$
+\end_inset
+
+ y 
+\begin_inset Formula $a\in D$
+\end_inset
+
+, la multiplicidad de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $P$
+\end_inset
+
+ es el menor 
+\begin_inset Formula $m\in\mathbb{N}_{0}$
+\end_inset
+
+ con 
+\begin_inset Formula $P^{(m)}(a)\neq0$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dado un anillo 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $A[X]$
+\end_inset
+
+ es un dominio euclídeo si y sólo si es un DIP, si y sólo si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $D$
+\end_inset
+
+ un dominio y 
+\begin_inset Formula $p\in D$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $p$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D$
+\end_inset
+
+ si y sólo si lo es en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+, lo es en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU, 
+\begin_inset Formula $p$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D$
+\end_inset
+
+ si y sólo si lo es en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+, si y sólo si es primo en 
+\begin_inset Formula $D$
+\end_inset
+
+, si y sólo si lo es en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $D$
+\end_inset
+
+ un DFU, definimos 
+\begin_inset Formula $\varphi:D\setminus0\to\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\varphi(a)$
+\end_inset
+
+ es el número de factores irreducibles en la factorización por irreducibles
+ de 
+\begin_inset Formula $a$
+\end_inset
+
+ en 
+\begin_inset Formula $D$
+\end_inset
+
+, contando repetidos, y para 
+\begin_inset Formula $a,b\in D\setminus\{0\}$
+\end_inset
+
+, 
+\begin_inset Formula $\varphi(ab)=\varphi(a)+\varphi(b)$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi(a)=0\iff a\in D^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU, 
+\begin_inset Formula $K$
+\end_inset
+
+ es su cuerpo de fracciones y 
+\begin_inset Formula $f\in D[X]$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+, es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+ [...] 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU si y sólo si lo es 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU y 
+\begin_inset Formula $K$
+\end_inset
+
+ es su cuerpo de fracciones, definimos la relación de equivalencia en 
+\begin_inset Formula $K$
+\end_inset
+
+ 
+\begin_inset Formula $x\sim y:\iff\exists u\in D^{*}:y=ux$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $[x]=xD^{*}$
+\end_inset
+
+ y, en particular, si 
+\begin_inset Formula $x\in D$
+\end_inset
+
+, 
+\begin_inset Formula $[x]$
+\end_inset
+
+ es el conjunto de los asociados de 
+\begin_inset Formula $x$
+\end_inset
+
+ en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ Definimos 
+\begin_inset Formula $\cdot:K\times(K/\sim)\to K/\sim$
+\end_inset
+
+ como 
+\begin_inset Formula $a(bD^{*})=(ab)D^{*}$
+\end_inset
+
+.
+ Esto está bien definido.
+ Además, 
+\begin_inset Formula $a(b(cD^{*}))=(ab)(cD^{*})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos 
+\begin_inset Formula $c:K[X]\to K/\sim$
+\end_inset
+
+ tal que, para 
+\begin_inset Formula $p\coloneqq\sum_{k\geq0}p_{k}X^{k}\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $c(p)\coloneqq\{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$
+\end_inset
+
+, y para 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+, si 
+\begin_inset Formula $a\in D\setminus\{0\}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $ap\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $c(p)\coloneqq a^{-1}c(ap)$
+\end_inset
+
+.
+ Esto está bien definido.
+ Si 
+\begin_inset Formula $c(p)=aD^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $a$
+\end_inset
+
+ es el 
+\series bold
+contenido
+\series default
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+ (
+\begin_inset Formula $a=c(p)$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $a\in K$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in K[X]$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $a\in D$
+\end_inset
+
+ y 
+\begin_inset Formula $p\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $a\mid p$
+\end_inset
+
+ en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $a\mid c(p)$
+\end_inset
+
+ en 
+\begin_inset Formula $D$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $c(ap)=ac(p)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $p\in D[X]\iff c(p)\in D$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un polinomio 
+\begin_inset Formula $p$
+\end_inset
+
+ es 
+\series bold
+primitivo
+\series default
+ si 
+\begin_inset Formula $c(p)=1$
+\end_inset
+
+, esto es, si 
+\begin_inset Formula $p\in D[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{mcd}_{k}p_{k}=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Lema de Gauss:
+\series default
+ Para 
+\begin_inset Formula $f,g\in D[X]$
+\end_inset
+
+, 
+\begin_inset Formula $c(fg)=c(f)c(g)$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $fg$
+\end_inset
+
+ es primitivo si y sólo si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ lo son.
+ [...]
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $f\in D[X]\setminus D$
+\end_inset
+
+ primitivo, 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ si y sólo si lo es en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall G,H\in K[X],(f=GH\implies\text{gr}(G)=0\lor\text{gr}(H)=0)$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $\forall g,h\in D[X],(f=gh\implies\text{gr}(g)=0\lor\text{gr}(h)=0)$
+\end_inset
+
+.
+ [...]
+\end_layout
+
+\begin_layout Standard
+De aquí que si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU con cuerpo de fracciones 
+\begin_inset Formula $K$
+\end_inset
+
+, los irreducibles de 
+\begin_inset Formula $D[X]$
+\end_inset
+
+ son precisamente los de 
+\begin_inset Formula $D$
+\end_inset
+
+ y los polinomios primitivos de 
+\begin_inset Formula $D[X]\setminus D$
+\end_inset
+
+ irreducibles en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+[...] Sean 
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo y 
+\begin_inset Formula $f\in K[X]$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(f)=1$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(f)>1$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene una raíz en 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ no es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{gr}(f)\in\{2,3\}$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+ si y sólo si no tiene raíces en 
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU con cuerpo de fracciones 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $f\coloneqq\sum_{k}a_{k}X^{k}\in D[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $n\coloneqq\text{gr}(f)$
+\end_inset
+
+, todas las raíces de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $K$
+\end_inset
+
+ son de la forma 
+\begin_inset Formula $\frac{r}{s}$
+\end_inset
+
+ con 
+\begin_inset Formula $r\mid a_{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $s\mid a_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterio de reducción:
+\series default
+ Sean 
+\begin_inset Formula $\phi:D\to K$
+\end_inset
+
+ un homomorfismo de anillos donde 
+\begin_inset Formula $D$
+\end_inset
+
+ es un DFU y 
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo, 
+\begin_inset Formula $\hat{\phi}:D[X]\to K[X]$
+\end_inset
+
+ el homomorfismo inducido por 
+\begin_inset Formula $\phi$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ un polinomio primitivo de 
+\begin_inset Formula $D[X]\setminus D$
+\end_inset
+
+, si 
+\begin_inset Formula $\hat{\phi}(f)$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(\hat{\phi}(f))=\text{gr}(f)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+En particular, si 
+\begin_inset Formula $p\in\mathbb{Z}$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $f\coloneqq\sum_{k}a_{k}X^{k}\in\mathbb{Z}[X]$
+\end_inset
+
+ es primitivo, 
+\begin_inset Formula $n\coloneqq\text{gr}(f)$
+\end_inset
+
+, 
+\begin_inset Formula $p\nmid a_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $\mathbb{Z}_{p}[X]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $\mathbb{Z}[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Criterio de Eisenstein:
+\series default
+ Sean 
+\begin_inset Formula $D$
+\end_inset
+
+ un DFU, 
+\begin_inset Formula $f\coloneqq\sum_{k}a_{k}X^{k}\in D[X]$
+\end_inset
+
+ primitivo y 
+\begin_inset Formula $n\coloneqq\text{gr}f$
+\end_inset
+
+, si existe un irreducible 
+\begin_inset Formula $p\in D$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall k\in\{0,\dots,n-1\},p\mid a_{k}$
+\end_inset
+
+ y 
+\begin_inset Formula $p^{2}\nmid a_{0}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es irreducible en 
+\begin_inset Formula $D[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $a\in\mathbb{Z}$
+\end_inset
+
+ y existe 
+\begin_inset Formula $p\in\mathbb{Z}$
+\end_inset
+
+ cuya multiplicidad en 
+\begin_inset Formula $a$
+\end_inset
+
+ es 1, 
+\begin_inset Formula $X^{n}-a$
+\end_inset
+
+ es irreducible.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $n\geq3$
+\end_inset
+
+, llamamos 
+\series bold
+raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas de la unidad
+\series default
+ o 
+\series bold
+de 1
+\series default
+ a las raíces de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+, que son los 
+\begin_inset Formula $n$
+\end_inset
+
+ vértices del 
+\begin_inset Formula $n$
+\end_inset
+
+-ágono regular inscrito en el círculo unidad de 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ con un vértice en el 1.
+ 
+\begin_inset Formula $X^{n}-1=(X-1)\Phi_{n}(X)$
+\end_inset
+
+, donde 
+\begin_inset Formula $\Phi_{n}(X)\coloneqq X^{n-1}+X^{n-2}+\dots+X+1$
+\end_inset
+
+ es el 
+\series bold
+
+\begin_inset Formula $n$
+\end_inset
+
+-ésimo polinomio ciclotómico
+\series default
+ y sus raíces en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ son las raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas de 1 distintas de 1.
+ En 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+, 
+\begin_inset Formula $X+1\mid\Phi_{4}(X)$
+\end_inset
+
+, pero si 
+\begin_inset Formula $n$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\Phi_{n}(X)$
+\end_inset
+
+ es irreducible.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Polinomios en varias indeterminadas
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{reminder}{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados un anillo conmutativo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+, definimos el 
+\series bold
+anillo de polinomios
+\series default
+ en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas con coeficientes en 
+\begin_inset Formula $A$
+\end_inset
+
+ como 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]\coloneqq A[X_{1},\dots,X_{n-1}][X_{n}]$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+indeterminadas
+\series default
+ a los símbolos 
+\begin_inset Formula $X_{1},\dots,X_{n}$
+\end_inset
+
+ y 
+\series bold
+polinomios en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas
+\series default
+ a los elementos de 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+.
+ Dados un anillo conmutativo 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ no es un cuerpo.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es un dominio si y sólo si lo es 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es un dominio, 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]^{*}=A^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es un DFU si y sólo si lo es 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ es un DIP si y sólo si 
+\begin_inset Formula $n=1$
+\end_inset
+
+ y 
+\begin_inset Formula $A$
+\end_inset
+
+ es un cuerpo.
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ e 
+\begin_inset Formula $i\coloneqq(i_{1},\dots,i_{n})\in\mathbb{N}^{n}$
+\end_inset
+
+, llamamos a 
+\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}\in A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ 
+\series bold
+monomio
+\series default
+ de 
+\series bold
+tipo
+\series default
+ 
+\begin_inset Formula $i$
+\end_inset
+
+ y coeficiente 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Todo 
+\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ se escribe de forma única como suma de monomios de distinto tipo,
+\begin_inset Formula 
+\[
+p:=\sum_{i\in\mathbb{N}^{n}}p_{i}X_{1}^{i_{1}}\cdots X_{n}^{i_{n}},
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $p_{i}=0$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $i\in\mathbb{N}^{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+PUAP en 
+\begin_inset Formula $n$
+\end_inset
+
+ indeterminadas:
+\series default
+ Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo conmutativo, 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+ y 
+\begin_inset Formula $u:A\to A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ la inclusión:
+\end_layout
+
+\begin_layout Enumerate
+Dados un homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, existe un único homomorfismo de anillos 
+\begin_inset Formula $\tilde{f}:A[X_{1},\dots,X_{n}]\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}\circ u=f$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}(X_{k})=b_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dados un anillo conmutativo 
+\begin_inset Formula $P$
+\end_inset
+
+, 
+\begin_inset Formula $T_{1},\dots,T_{n}\in P$
+\end_inset
+
+ y un homomorfismo 
+\begin_inset Formula $v:A\to P$
+\end_inset
+
+ tales que, dados un homomorfismo de anillos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, existe un único homomorfismo 
+\begin_inset Formula $\tilde{f}:P\to B$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\tilde{f}\circ v=f$
+\end_inset
+
+ y 
+\begin_inset Formula $\tilde{f}(T_{k})=b_{k}$
+\end_inset
+
+ para 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+, existe un isomorfismo 
+\begin_inset Formula $\phi:A[X_{1},\dots,X_{n}]\to P$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\phi\circ u=v$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi(X_{k})=T_{k}$
+\end_inset
+
+ para cada 
+\begin_inset Formula $k\in\{1,\dots,n\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así:
+\end_layout
+
+\begin_layout Enumerate
+Dados dos anillos conmutativos 
+\begin_inset Formula $A\subseteq B$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{1},\dots,b_{n}\in B$
+\end_inset
+
+, el 
+\series bold
+homomorfismo de sustitución
+\series default
+ 
+\begin_inset Formula $S:A[X_{1},\dots,X_{n}]\to B$
+\end_inset
+
+ viene dado por 
+\begin_inset Formula $p(b_{1},\dots,b_{n})\coloneqq S(p)\coloneqq\sum_{i\in\mathbb{N}^{n}}p_{i}b_{1}^{i_{1}}\cdots b_{n}^{i_{n}}$
+\end_inset
+
+.
+ Su imagen es el subanillo de 
+\begin_inset Formula $B$
+\end_inset
+
+ generado por 
+\begin_inset Formula $A\cup\{b_{1},\dots,b_{n}\}$
+\end_inset
+
+, 
+\begin_inset Formula $A[b_{1},\dots,b_{n}]$
+\end_inset
+
+, y dados dos homomorfismos de anillos 
+\begin_inset Formula $f,g:A[b_{1},\dots,b_{n}]\to C$
+\end_inset
+
+, 
+\begin_inset Formula $f=g$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $f|_{A}=g|_{A}$
+\end_inset
+
+ y 
+\begin_inset Formula $f(b_{k})=g(b_{k})$
+\end_inset
+
+ para todo 
+\begin_inset Formula $k$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean 
+\begin_inset Formula $A$
+\end_inset
+
+ un anillo y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ una permutación de 
+\begin_inset Formula $\mathbb{N}_{n}$
+\end_inset
+
+ con inversa 
+\begin_inset Formula $\tau\coloneqq\sigma^{-1}$
+\end_inset
+
+, tomando 
+\begin_inset Formula $B=A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ y 
+\begin_inset Formula $b_{k}=X_{\sigma(k)}$
+\end_inset
+
+ en el punto anterior obtenemos un automorfismo 
+\begin_inset Formula $\hat{\sigma}$
+\end_inset
+
+ en 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+ con inversa 
+\begin_inset Formula $\hat{\tau}$
+\end_inset
+
+ que permuta las indeterminadas.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A[X_{1},\dots,X_{n},Y_{1},\dots,Y_{m}]\cong A[X_{1},\dots,X_{n}][Y_{1},\dots,Y_{m}]\cong A[Y_{1},\dots,Y_{m}][X_{1},\dots,X_{n}]$
+\end_inset
+
+, por lo que en la práctica no distinguimos entre estos anillos.
+\end_layout
+
+\begin_layout Enumerate
+Todo homomorfismo de anillos conmutativos 
+\begin_inset Formula $f:A\to B$
+\end_inset
+
+ induce un homomorfismo 
+\begin_inset Formula $\hat{f}:A[X_{1},\dots,X_{n}]\to B[X_{1},\dots,X_{n}]$
+\end_inset
+
+ dado por 
+\begin_inset Formula $\hat{f}(p)\coloneqq\sum_{i\in\mathbb{N}^{n}}f(p_{i})X_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+grado
+\series default
+ de un monomio 
+\begin_inset Formula $aX_{1}^{i_{1}}\cdots X_{n}^{i_{n}}$
+\end_inset
+
+ a 
+\begin_inset Formula $i_{1}+\dots+i_{n}$
+\end_inset
+
+, y grado de 
+\begin_inset Formula $p\in A[X_{1},\dots,X_{n}]\setminus0$
+\end_inset
+
+, 
+\begin_inset Formula $\text{gr}(p)$
+\end_inset
+
+, al mayor de los grados de los monomios no nulos en la expresión por monomios
+ de 
+\begin_inset Formula $p$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\text{gr}(p+q)\leq\max\{\text{gr}(p),\text{gr}(q)\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(pq)\leq\text{gr}(p)+\text{gr}(q)$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Un polinomio es 
+\series bold
+homogéneo
+\series default
+ de grado 
+\begin_inset Formula $n$
+\end_inset
+
+ si es suma de monomios de grado 
+\begin_inset Formula $n$
+\end_inset
+
+.
+ Todo polinomio se escribe de modo único como suma de polinomios homogéneos
+ de distintos grados, sin más que agrupar los monomios de igual grado en
+ la expresión como suma de monomios.
+ Así, si 
+\begin_inset Formula $D$
+\end_inset
+
+ es un dominio, 
+\begin_inset Formula $\text{gr}(pq)=\text{gr}(p)+\text{gr}(q)$
+\end_inset
+
+ para cualesquiera 
+\begin_inset Formula $p,q\in D[X_{1},\dots,X_{n}]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{reminder}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/ac/nc.lyx b/ac/nc.lyx
new file mode 100644
index 0000000..cfcca89
--- /dev/null
+++ b/ac/nc.lyx
@@ -0,0 +1,152 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input{../defs}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+kern-1em
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+\binom{n}{k} & =\binom{n}{n-k}; &  &  & \binom{r}{k} & =(-1)^{k}\binom{k-r-1}{k};\\
+\binom{r}{k} & =\frac{r}{k}\binom{r-1}{k-1}, & k & \neq0; & \binom{n}{m} & =(-1)^{n-m}\binom{-(m+1)}{n-m}, & n & \geq0;\\
+\binom{r}{k} & =\frac{r}{r-k}\binom{r-1}{k}, & k & \neq r; & \sum_{k=0}^{n}\binom{r+k}{k} & =\binom{r+n+1}{n}, & n & \geq0;\\
+\binom{r}{k} & =\binom{r-1}{k}+\binom{r-1}{k-1}; &  &  & \sum_{k=0}^{n}\binom{k}{m} & =\binom{n+1}{m+1}, & m,n & \geq0;
+\end{align*}
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+\binom{r}{m}\binom{m}{k} & =\binom{r}{k}\binom{r-k}{m-k}, & \sum_{k}\binom{r}{k}\binom{s}{n-k} & =\binom{r+s}{n};\\
+\sum_{k}\binom{r}{m+k}\binom{s}{n+k} & =\binom{r+s}{r-m+n}, & \sum_{k}\binom{r}{k}\binom{s+k}{n}(-1)^{r-k} & =\binom{s}{n-r}, & r & \geq0;
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+\sum_{k=0}^{r}\binom{r-k}{m}\binom{s}{k-t}(-1)^{k-t} & =\binom{r-t-s}{r-t-m}, & t,r,m & \geq0;\\
+\sum_{k=0}^{r}\binom{r-k}{m}\binom{s+k}{n} & =\binom{r+s+1}{m+n+1}, & n\geq s & \geq0,\ m,r\geq0;\\
+\sum_{k\geq0}\binom{r-tk}{k}\binom{s-t(n-k)}{n-k}\frac{r}{r-tk} & =\binom{r+s-tn}{n};
+\end{align*}
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+\sum_{k}\binom{n}{k}x(x-kz)^{k-1}(y+kz)^{n-k} & =(x+y)^{n}, & x & \neq0;
+\end{align*}
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+\sum_{k}\binom{r}{k}x^{k}y^{r-k} & =(x+y)^{r}, & r & \geq0; & \sum_{k}\binom{r}{k}x^{k} & =(1+x)^{r}, & r & \geq0\text{ o }|x|<1;
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document