x

2 files changed, 653 insertions, 56 deletions

diff --git a/ealg/n1.lyx b/ealg/n1.lyx
index 7434dd6..6086c51 100644
--- a/ealg/n1.lyx
+++ b/ealg/n1.lyx
@@ -2329,6 +2329,22 @@ donde
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hbox{
+\backslash
+vline
+\backslash
+vbox{
+\end_layout
+
+\end_inset
+
 Para 
 \begin_inset Formula $n,m\in\mathbb{Z}^{+}$
 \end_inset
@@ -2338,6 +2354,17 @@ Para
 \end_inset
 
 .
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Section
@@ -2663,6 +2690,223 @@ primitiva
 .
 \end_layout
 
+\begin_layout Standard
+Dado 
+\begin_inset Formula $f:=Y^{3}+3pY+2q\in\mathbb{C}[X]$
+\end_inset
+
+, si 
+\begin_inset Formula $\omega:=e^{2\pi i/3}$
+\end_inset
+
+, existe 
+\begin_inset Formula $k\in\{0,1,2\}$
+\end_inset
+
+ tal que, si
+\begin_inset Formula 
+\begin{align*}
+r & :=\sqrt[3]{-q+\sqrt{q^{2}+p^{3}}}, & s & :=\omega^{k}\sqrt[3]{-q-\sqrt{q^{2}-p^{3}}},
+\end{align*}
+
+\end_inset
+
+las raíces de 
+\begin_inset Formula $f$
+\end_inset
+
+ son 
+\begin_inset Formula $r+s$
+\end_inset
+
+, 
+\begin_inset Formula $r\omega+s\omega^{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $r\omega^{2}+s\omega$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Sean 
+\begin_inset Formula $r$
+\end_inset
+
+ y 
+\begin_inset Formula $s$
+\end_inset
+
+ incógnitas tales que 
+\begin_inset Formula $Y=r+s$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+f(Y)=r^{3}+3(r+s)rs+s^{3}+3p(r+s)+2q=(r^{3}+s^{3}+2q)+3(r+s)(rs+p).
+\]
+
+\end_inset
+
+Podemos tomar 
+\begin_inset Formula $rs=-p$
+\end_inset
+
+, ya que como 
+\begin_inset Formula $r=Y-s$
+\end_inset
+
+ esto equivale a que 
+\begin_inset Formula $(Y-s)s=Ys-s^{2}=-p$
+\end_inset
+
+, una ecuación de segundo grado con incógnita 
+\begin_inset Formula $s$
+\end_inset
+
+ que tiene sus raíces en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $rs+p=0$
+\end_inset
+
+ y 
+\begin_inset Formula $f(Y)=r^{3}+s^{3}+2q=0$
+\end_inset
+
+.
+ Multiplicando esto por 
+\begin_inset Formula $r^{3}$
+\end_inset
+
+ o 
+\begin_inset Formula $s^{3}$
+\end_inset
+
+, 
+\begin_inset Formula $r^{6}+2qr^{3}-p^{3}=0$
+\end_inset
+
+ y 
+\begin_inset Formula $s^{6}+2qs^{3}-p^{3}=0$
+\end_inset
+
+, luego 
+\begin_inset Formula $r$
+\end_inset
+
+ y 
+\begin_inset Formula $s$
+\end_inset
+
+ son raíces de 
+\begin_inset Formula $T^{6}+2qT^{3}-p^{3}=0$
+\end_inset
+
+, con soluciones 
+\begin_inset Formula $T^{3}=-q\pm\sqrt{q^{2}+p^{3}}$
+\end_inset
+
+.
+ Podemos suponer 
+\begin_inset Formula $r^{3}=-q+\sqrt{q^{2}+p^{3}}$
+\end_inset
+
+ y 
+\begin_inset Formula $s^{3}=-q-\sqrt{q^{2}+p^{3}}$
+\end_inset
+
+, y entonces existen 
+\begin_inset Formula $i,j\in\{0,1,2\}$
+\end_inset
+
+ con
+\begin_inset Formula 
+\begin{align*}
+r & =\omega^{i}\sqrt[3]{-q+\sqrt{q^{2}+p^{3}}}, & s & =\omega^{j}\sqrt[3]{-q-\sqrt{q^{2}+p^{3}}}.
+\end{align*}
+
+\end_inset
+
+Para 
+\begin_inset Formula $a,b\in\mathbb{Z}$
+\end_inset
+
+, 
+\begin_inset Formula $(r\omega^{a})(s\omega^{b})=-p\omega^{a+b}$
+\end_inset
+
+, de modo que las únicas tres raíces posibles son aquellas en las que 
+\begin_inset Formula $\omega^{a+b}=1$
+\end_inset
+
+, que son 
+\begin_inset Formula $r+s$
+\end_inset
+
+, 
+\begin_inset Formula $r\omega+s\omega^{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $r\omega^{2}+s\omega$
+\end_inset
+
+, y con esto podemos elegir 
+\begin_inset Formula $r$
+\end_inset
+
+ tal que 
+\begin_inset Formula $i=0$
+\end_inset
+
+ y habrá un único 
+\begin_inset Formula $j\in\{0,1,2\}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $r+s$
+\end_inset
+
+ es raíz.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:=aX^{3}+bX^{2}+cX+d\in\mathbb{C}[X]$
+\end_inset
+
+, podemos obtener las raíces de 
+\begin_inset Formula $f(X)$
+\end_inset
+
+ obteniendo las de 
+\begin_inset Formula $(\frac{1}{a}f)(X-\frac{b}{3a})$
+\end_inset
+
+, que será de la forma 
+\begin_inset Formula $X^{3}+3pX+2q$
+\end_inset
+
+.
+\end_layout
+
 \begin_layout Section
 Polinomios en varias variables
 \end_layout
@@ -3404,10 +3648,14 @@ y son simétricos.
 
 .
  Si 
-\begin_inset Formula $\tilde{s}_{1}(X_{1},\dots,X_{n-1}),\dots,\tilde{s}_{n-1}(X_{1},\dots,X_{n-1})$
+\begin_inset Formula 
+\[
+\tilde{s}_{1}(X_{1},\dots,X_{n-1}),\dots,\tilde{s}_{n-1}(X_{1},\dots,X_{n-1})
+\]
+
 \end_inset
 
- son los polinomios simétricos elementales en las variables 
+son los polinomios simétricos elementales en las variables 
 \begin_inset Formula $X_{1},\dots,X_{n-1}$
 \end_inset
 
@@ -3427,36 +3675,7 @@ y son simétricos.
 \end_layout
 
 \begin_layout Standard
-
-\series bold
-Teorema fundamental de los polinomios simétricos:
-\series default
- Sea 
-\begin_inset Formula $S[X_{1},\dots,X_{n}]$
-\end_inset
-
- el subanillo de los polinomios simétricos de 
-\begin_inset Formula $A[X_{1},\dots,X_{n}]$
-\end_inset
-
-, el homomorfismo de evaluación 
-\begin_inset Formula $\varphi:A[X_{1},\dots,X_{n}]\to S[X_{1},\dots,X_{n}]$
-\end_inset
-
- con 
-\begin_inset Formula $\varphi(X_{i})=s_{i}$
-\end_inset
-
- es un isomorfismo, es decir, todo polinomio simétrico se escribe de forma
- única como expresión polinómica en los polinomios simétricos elementales.
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Fórmulas de Cardano-Vieta:
-\series default
- Sean 
+Sean 
 \begin_inset Formula $A$
 \end_inset
 
@@ -3488,6 +3707,11 @@ Fórmulas de Cardano-Vieta:
 
 \end_inset
 
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
 En efecto, sea 
 \begin_inset Formula 
 \[
@@ -3507,6 +3731,36 @@ f(X)=(X-\alpha_{1})\cdots(X-\alpha_{n})=f(\alpha_{1},\dots,\alpha_{n},X)=X^{n}+s
 
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Teorema fundamental de los polinomios simétricos:
+\series default
+ Sea 
+\begin_inset Formula $S[X_{1},\dots,X_{n}]$
+\end_inset
+
+ el subanillo de los polinomios simétricos de 
+\begin_inset Formula $A[X_{1},\dots,X_{n}]$
+\end_inset
+
+, el homomorfismo de evaluación 
+\begin_inset Formula $\varphi:A[X_{1},\dots,X_{n}]\to S[X_{1},\dots,X_{n}]$
+\end_inset
+
+ con 
+\begin_inset Formula $\varphi(X_{i})=s_{i}$
+\end_inset
+
+ es un isomorfismo, es decir, todo polinomio simétrico se escribe de forma
+ única como expresión polinómica en los polinomios simétricos elementales.
+\end_layout
+
 \begin_layout Standard
 El 
 \series bold
diff --git a/ealg/n2.lyx b/ealg/n2.lyx
index 49dbf78..3d07f3b 100644
--- a/ealg/n2.lyx
+++ b/ealg/n2.lyx
@@ -237,6 +237,54 @@ Algunas extensiones son
 \end_inset
 
 .
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hbox{
+\backslash
+vline
+\backslash
+vbox{
+\end_layout
+
+\end_inset
+
+Dados 
+\begin_inset Formula $c,d\in\mathbb{Z}$
+\end_inset
+
+ no cuadrados, 
+\begin_inset Formula $\mathbb{Q}(\sqrt{c})=\mathbb{Q}(\sqrt{d})$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $cd$
+\end_inset
+
+ es un cuadrado en 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -1184,6 +1232,69 @@ Dadas
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hbox{
+\backslash
+vline
+\backslash
+vbox{
+\end_layout
+
+\end_inset
+
+Dado un homomorfismo de cuerpos 
+\begin_inset Formula $f:K\to L$
+\end_inset
+
+, 
+\begin_inset Formula $K$
+\end_inset
+
+ y 
+\begin_inset Formula $L$
+\end_inset
+
+ tienen un mismo subcuerpo primo 
+\begin_inset Formula $P$
+\end_inset
+
+ (
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+) y 
+\begin_inset Formula $f$
+\end_inset
+
+ es un 
+\begin_inset Formula $P$
+\end_inset
+
+-encaje.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Si 
 \begin_inset Formula $K\subseteq L$
 \end_inset
@@ -2539,6 +2650,116 @@ Si
 \end_layout
 
 \begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+vspace{-1ex}
+\backslash
+hbox{
+\backslash
+vline
+\backslash
+vbox{
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+4.
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $f\in K[X]$
+\end_inset
+
+ es irreducible de grado al menos 2, 
+\begin_inset Formula $f$
+\end_inset
+
+ no tiene raíces en ninguna extensión finita 
+\begin_inset Formula $L$
+\end_inset
+
+ de 
+\begin_inset Formula $K$
+\end_inset
+
+ con 
+\begin_inset Formula $[L:K]$
+\end_inset
+
+ coprimo con 
+\begin_inset Formula $\text{gr}f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+5.
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $X^{n}-a\in K[X]$
+\end_inset
+
+ es irreducible, 
+\begin_inset Formula $\beta$
+\end_inset
+
+ una raíz de 
+\begin_inset Formula $X^{n}-a$
+\end_inset
+
+ en una extensión de 
+\begin_inset Formula $K$
+\end_inset
+
+ y 
+\begin_inset Formula $m\mid n$
+\end_inset
+
+, 
+\begin_inset Formula $[K(\beta^{m}):K]=n/m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
 Sea 
 \begin_inset Formula $K\subseteq L$
 \end_inset
@@ -3559,33 +3780,20 @@ Dada una familia
  [...] con el producto componente a componente.
 \end_layout
 
-\begin_layout Enumerate
-\begin_inset Argument item:1
+\begin_layout Standard
+\begin_inset ERT
 status open
 
 \begin_layout Plain Layout
-4.
-\end_layout
 
-\end_inset
 
-Llamamos 
-\series bold
-grupo cíclico
-\series default
- de orden 
-\begin_inset Formula $n\in\mathbb{N}^{*}$
-\end_inset
+\backslash
+eremember
+\end_layout
 
- a 
-\begin_inset Formula $C_{n}:=\{1,a,a^{2},\dots,a^{n-1}\}$
 \end_inset
 
- con [...] 
-\begin_inset Formula $a^{i}a^{j}:=a^{[i+j]_{n}}$
-\end_inset
 
- [...].
 \end_layout
 
 \begin_layout Standard
@@ -3596,7 +3804,7 @@ status open
 
 
 \backslash
-eremember
+sremember{GyA}
 \end_layout
 
 \end_inset
@@ -3604,20 +3812,33 @@ eremember
 
 \end_layout
 
-\begin_layout Standard
-\begin_inset ERT
+\begin_layout Enumerate
+\begin_inset Argument item:1
 status open
 
 \begin_layout Plain Layout
+4.
+\end_layout
 
+\end_inset
 
-\backslash
-sremember{GyA}
-\end_layout
+Llamamos 
+\series bold
+grupo cíclico
+\series default
+ de orden 
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
 
+ a 
+\begin_inset Formula $C_{n}:=\{1,a,a^{2},\dots,a^{n-1}\}$
 \end_inset
 
+ con [...] 
+\begin_inset Formula $a^{i}a^{j}:=a^{[i+j]_{n}}$
+\end_inset
 
+ [...].
 \end_layout
 
 \begin_layout Enumerate
@@ -4738,5 +4959,127 @@ Equivale a ser algebraica y finitamente generada.
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hbox{
+\backslash
+vline
+\backslash
+vbox{
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Además, dada una extensión 
+\begin_inset Formula $K\subseteq F$
+\end_inset
+
+ con cuerpos intermedios 
+\begin_inset Formula $L$
+\end_inset
+
+ y 
+\begin_inset Formula $M$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $[LM:K]$
+\end_inset
+
+ es finito si y sólo si lo son 
+\begin_inset Formula $[L:K]$
+\end_inset
+
+ y 
+\begin_inset Formula $[M:K]$
+\end_inset
+
+, en cuyo caso 
+\begin_inset Formula $[L:K],[M:K]\mid[LM:K]$
+\end_inset
+
+ y 
+\begin_inset Formula $[LM:K]\leq[L:K][M:K]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $L$
+\end_inset
+
+ y 
+\begin_inset Formula $M$
+\end_inset
+
+ son extensiones algebraicas de 
+\begin_inset Formula $K$
+\end_inset
+
+, también lo es 
+\begin_inset Formula $LM$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $[LM:K]=[L:K][M:K]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $L\cap M=K$
+\end_inset
+
+.
+ El recíproco no se cumple.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $[L:K]\leq2$
+\end_inset
+
+ y 
+\begin_inset Formula $L\cap M=K$
+\end_inset
+
+, entonces 
+\begin_inset Formula $[LM:K]=[L:K][M:K]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document