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| -rw-r--r-- | ac/n.lyx | 82 | ||||
| -rw-r--r-- | ac/n1.lyx | 2773 | ||||
| -rw-r--r-- | ac/n3.lyx | 1442 | ||||
| -rw-r--r-- | ac/n4.lyx | 3683 | ||||
| -rw-r--r-- | ac/n5.lyx | 4379 | ||||
| -rw-r--r-- | ac/na.lyx | 1250 | ||||
| -rw-r--r-- | ac/nb.lyx | 2735 | ||||
| -rw-r--r-- | ac/nc.lyx | 152 |
8 files changed, 12077 insertions, 4419 deletions
@@ -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 @@ -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 @@ -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 @@ -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 |