x

1 files changed, 367 insertions, 24 deletions

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