1 files changed, 343 insertions, 0 deletions
diff --git a/ealg/n4.lyx b/ealg/n4.lyx
index d6843c6..9673c48 100644
--- a/ealg/n4.lyx
+++ b/ealg/n4.lyx
@@ -921,6 +921,131 @@ Demostración:
 
 \end_layout
 
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hbox{
+\backslash
+vline
+\backslash
+hspace{2pt}
+\backslash
+vbox{
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ es una extensión de grado 2, 
+\begin_inset Formula $L$
+\end_inset
+
+ es el cuerpo de descomposición sobre 
+\begin_inset Formula $K$
+\end_inset
+
+ de un polinomio de 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+ Sean 
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ y 
+\begin_inset Formula $K\subseteq M$
+\end_inset
+
+ son extensiones admisibles:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $L$
+\end_inset
+
+ es un cuerpo de descomposición de 
+\begin_inset Formula $f\in K[X]$
+\end_inset
+
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $LM$
+\end_inset
+
+ es un cuerpo de descomposición de 
+\begin_inset Formula $f$
+\end_inset
+
+ sobre 
+\begin_inset Formula $M$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $L$
+\end_inset
+
+ y 
+\begin_inset Formula $M$
+\end_inset
+
+ son cuerpos de descomposición respectivos de 
+\begin_inset Formula $f,g\in K[X]$
+\end_inset
+
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $LM$
+\end_inset
+
+ es un cuerpo de descomposición de 
+\begin_inset Formula $fg$
+\end_inset
+
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Grupo de Galois de un polinomio
 \end_layout
@@ -2352,5 +2477,223 @@ Sean
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hbox{
+\backslash
+vline
+\backslash
+hspace{2pt}
+\backslash
+vbox{
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo:
+\end_layout
+
+\begin_layout Enumerate
+Todo 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ algebraico sobre 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $\text{Irr}(\alpha,\mathbb{Z}_{p})=\text{Irr}(\alpha^{p},\mathbb{Z}_{p})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La función 
+\begin_inset Formula $h:\mathbb{F}_{p^{n}}\to\mathbb{F}_{p^{n}}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $h(x):=x^{p}$
+\end_inset
+
+ es biyectiva.
+\end_layout
+
+\begin_layout Enumerate
+La suma de todos los elementos de un cuerpo finito con más de dos elementos
+ es 0.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $L$
+\end_inset
+
+ es el cuerpo de descomposición sobre 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ de un 
+\begin_inset Formula $f\in\mathbb{Z}_{p}[X]$
+\end_inset
+
+ que se factoriza en irreducibles como 
+\begin_inset Formula $f=:f_{1}\cdots f_{r}$
+\end_inset
+
+, sea 
+\begin_inset Formula $[L:\mathbb{Z}_{p}]=\text{mcm}\{\text{gr}f_{1},\dots,\text{gr}f_{r}\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $p=2k+1$
+\end_inset
+
+ es un primo impar, llamamos 
+\series bold
+restos cuadráticos
+\series default
+ módulo 
+\begin_inset Formula $p$
+\end_inset
+
+ a los cuadrados no nulos en 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+, 
+\begin_inset Formula $1^{2},2^{2},\dots,k^{2}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\prod_{i=1}^{k}i^{2}=(-1)^{k+1}$
+\end_inset
+
+ y, si 
+\begin_inset Formula $p\neq3$
+\end_inset
+
+, 
+\begin_inset Formula $\sum_{i=1}^{k}i^{k}=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Algunos grupos finitos:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{F}_{4}=\{a\alpha+b\}_{a,b\in\mathbb{Z}_{2}}$
+\end_inset
+
+ se obtiene al añadir a 
+\begin_inset Formula $\mathbb{Z}_{2}$
+\end_inset
+
+ una raíz 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ del irreducible 
+\begin_inset Formula $X^{2}+X+1\in\mathbb{Z}_{2}[X]$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\mathbb{F}_{4}^{*}=\langle\alpha\rangle=\langle\alpha+1\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{F}_{8}=\{a\beta^{2}+b\beta+c\}_{a,b,c\in\mathbb{Z}_{2}}$
+\end_inset
+
+ se obtiene al añadir a 
+\begin_inset Formula $\mathbb{Z}_{2}$
+\end_inset
+
+ una raíz 
+\begin_inset Formula $\beta$
+\end_inset
+
+ del irreducible 
+\begin_inset Formula $X^{3}+X+1\in\mathbb{Z}_{2}[X]$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\mathbb{F}_{8}^{*}=\langle x\rangle$
+\end_inset
+
+ para 
+\begin_inset Formula $x\in\mathbb{F}_{8}^{*}\setminus\{1\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{F}_{9}=\{a\gamma+b\}_{a,b\in\mathbb{Z}_{3}}$
+\end_inset
+
+ se obtiene al añadir a 
+\begin_inset Formula $\mathbb{Z}_{3}$
+\end_inset
+
+ una raíz 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ del irreducible 
+\begin_inset Formula $X^{2}+1\in\mathbb{Z}_{3}[X]$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\mathbb{F}_{9}^{*}=\langle\gamma+1\rangle$
+\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