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-rw-r--r--ealg/n.lyx14
-rw-r--r--ealg/n2.lyx304
-rw-r--r--ealg/n4.lyx1233
3 files changed, 1544 insertions, 7 deletions
diff --git a/ealg/n.lyx b/ealg/n.lyx
index 12db22d..099164a 100644
--- a/ealg/n.lyx
+++ b/ealg/n.lyx
@@ -176,5 +176,19 @@ filename "n2.lyx"
\end_layout
+\begin_layout Chapter
+Cuerpos de descomposición
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n4.lyx"
+
+\end_inset
+
+
+\end_layout
+
\end_body
\end_document
diff --git a/ealg/n2.lyx b/ealg/n2.lyx
index b2edddd..49dbf78 100644
--- a/ealg/n2.lyx
+++ b/ealg/n2.lyx
@@ -3728,8 +3728,210 @@ grupo de Klein
\end_inset
.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Necesario para las demostraciones comentadas.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Dado un grupo
+\begin_inset Formula $G$
+\end_inset
+
+, llamamos
+\series bold
+exponente
+\series default
+ o
+\series bold
+periodo
+\series default
+ de
+\begin_inset Formula $G$
+\end_inset
+
+,
+\begin_inset Formula $\text{Exp}(G)$
+\end_inset
+
+, al menor
+\begin_inset Formula $n\in\mathbb{N}^{*}$
+\end_inset
+
+ tal que
+\begin_inset Formula $\forall g\in G,g^{n}=1$
+\end_inset
+
+, o a
+\begin_inset Formula $\infty$
+\end_inset
+
+ si este no existe.
+ [...] Si un grupo es finito tiene periodo finito [...].
+\end_layout
+
+\begin_layout Plain Layout
+[...] Una
+\series bold
+descomposición primaria
+\series default
+ o
+\series bold
+indescomponible
+\series default
+ de un grupo abeliano finito
+\begin_inset Formula $A$
+\end_inset
+
+ es una expresión de la forma
+\begin_inset Formula
+\begin{align*}
+A= & \langle a_{11}\rangle_{p_{1}^{\alpha_{11}}}\oplus\dots\oplus\langle a_{1m_{1}}\rangle_{p_{1}^{\alpha_{1m_{1}}}}\oplus\\
+ & \dots\oplus\\
+ & \langle a_{k1}\rangle_{p_{k}^{\alpha_{k1}}}\oplus\dots\oplus\langle a_{km_{k}}\rangle_{p_{k}^{\alpha_{km_{k}}}},
+\end{align*}
+
+\end_inset
+
+donde
+\begin_inset Formula $p_{1}<\dots<p_{k}$
+\end_inset
+
+ son los primos que dividen a
+\begin_inset Formula $|A|$
+\end_inset
+
+ y
+\begin_inset Formula $\alpha_{i1}\geq\dots\geq\alpha_{im_{i}}\geq1$
+\end_inset
+
+ para cada
+\begin_inset Formula $i\in\{1,\dots,k\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+Como
+\series bold
+teorema
+\series default
+, todo grupo abeliano tiene una descomposición primaria [...].
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+ Para todo grupo finito
+\begin_inset Formula $G$
+\end_inset
+
+,
+\begin_inset Formula $\text{Exp}G\mid|G|$
+\end_inset
+
+, y en particular, para
+\begin_inset Formula $g\in G$
+\end_inset
+
+,
+\begin_inset Formula $g^{|G|}=1$
+\end_inset
+
+.
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto,
+\begin_inset Formula $\text{Exp}G=\text{mcm}_{g\in G}|g|$
+\end_inset
+
+, pero como para todo
+\begin_inset Formula $g\in G$
+\end_inset
+
+,
+\begin_inset Formula $\langle g\rangle$
+\end_inset
+
+ es un subgrupo de
+\begin_inset Formula $G$
+\end_inset
+
+,
+\begin_inset Formula $|g|\mid|G|$
+\end_inset
+
+, luego
+\begin_inset Formula $|G|$
+\end_inset
+
+ es múltiplo de todo
+\begin_inset Formula $|g|$
+\end_inset
+
+ y por tanto lo es de
+\begin_inset Formula $\text{Exp}G$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
Dado un cuerpo
-\begin_inset Formula $K$
+\begin_inset Formula $K\neq0$
\end_inset
, todo subgrupo finito de
@@ -3746,6 +3948,98 @@ grupo de Klein
\end_inset
es cíclico.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+Llamando
+\begin_inset Formula $m:=\text{Exp}K^{*}$
+\end_inset
+
+,
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ está formado por raíces de
+\begin_inset Formula $X^{m}-1$
+\end_inset
+
+, de modo que
+\begin_inset Formula $|K^{*}|\leq m$
+\end_inset
+
+ y, como
+\begin_inset Formula $m\mid|K^{*}|>0$
+\end_inset
+
+ (
+\begin_inset Formula $1\in K^{*}$
+\end_inset
+
+),
+\begin_inset Formula $m=|K^{*}|$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ tiene una descomposición primaria de la forma
+\begin_inset Formula $\langle a_{1}\rangle_{p_{1}^{\alpha_{1}}}\oplus\dots\oplus\langle a_{k}\rangle_{p_{k}^{\alpha_{k}}}$
+\end_inset
+
+ con cada
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ distinto, pues si su descomposición primaria es
+\begin_inset Formula $\bigoplus_{i=1}^{k}\bigoplus_{j=1}^{m_{i}}\langle a_{ij}\rangle_{p_{i}^{\alpha_{ij}}}$
+\end_inset
+
+, el orden es
+\begin_inset Formula $\prod_{i,j}p_{i}^{\alpha_{ij}}$
+\end_inset
+
+ y el exponente es
+\begin_inset Formula $\prod_{i}p_{i}^{\alpha_{i1}}$
+\end_inset
+
+, dado que
+\begin_inset Formula $p_{i}^{\alpha_{ij}}\mid p_{i}^{\alpha_{i1}}$
+\end_inset
+
+ para todo
+\begin_inset Formula $j$
+\end_inset
+
+, y para que estos coincidan debe ser cada
+\begin_inset Formula $m_{i}=1$
+\end_inset
+
+.
+ Entonces el orden de
+\begin_inset Formula $a:=a_{1}+\dots+a_{k}$
+\end_inset
+
+ es
+\begin_inset Formula $m$
+\end_inset
+
+, luego
+\begin_inset Formula $\langle a\rangle=K^{*}$
+\end_inset
+
+ es cíclico.
+\end_layout
+
+\end_inset
+
+
\end_layout
\begin_layout Standard
@@ -4244,7 +4538,7 @@ status open
\begin_inset Formula $\alpha\in M$
\end_inset
- y por tanto
+ y
\begin_inset Formula $\alpha$
\end_inset
@@ -4345,10 +4639,6 @@ Para
\end_layout
\end_deeper
-\begin_layout Enumerate
-Ser finitamente generada.
-\end_layout
-
\begin_layout Standard
Una propiedad relativa a extensiones es
\series bold
@@ -4426,7 +4716,7 @@ LM=LK(\alpha_{1},\dots,\alpha_{n})=L(\alpha_{1},\dots,\alpha_{m}),
\end_inset
- pues
+pues
\begin_inset Formula $LK(\alpha_{1},\dots,\alpha_{n})$
\end_inset
diff --git a/ealg/n4.lyx b/ealg/n4.lyx
new file mode 100644
index 0000000..ffcd066
--- /dev/null
+++ b/ealg/n4.lyx
@@ -0,0 +1,1233 @@
+#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
+Sea
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ una extensión de cuerpos,
+\begin_inset Formula $f\in K[X]$
+\end_inset
+
+ de grado
+\begin_inset Formula $n\geq0$
+\end_inset
+
+ se
+\series bold
+descompone
+\series default
+ o
+\series bold
+factoriza completamente
+\series default
+ en
+\begin_inset Formula $L$
+\end_inset
+
+ si existen
+\begin_inset Formula $c\in K$
+\end_inset
+
+ y
+\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}\in L$
+\end_inset
+
+ con
+\begin_inset Formula
+\[
+f(X)=c\prod_{i=1}^{n}(X-\alpha_{i}).
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean
+\begin_inset Formula $K$
+\end_inset
+
+ un cuerpo y
+\begin_inset Formula ${\cal P}\subseteq K[X]\setminus0$
+\end_inset
+
+, un
+\series bold
+cuerpo de descomposición
+\series default
+ de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+ es una extensión
+\begin_inset Formula $L$
+\end_inset
+
+ de
+\begin_inset Formula $K$
+\end_inset
+
+ en la que todos los polinomios de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ se descomponen completamente y sus raíces generan
+\begin_inset Formula $L$
+\end_inset
+
+.
+ Un cuerpo de descomposición de
+\begin_inset Formula $f\in K[X]\setminus0$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+ es uno de
+\begin_inset Formula $\{f\}$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ una extensión de cuerpos,
+\begin_inset Formula $f\in K[X]\setminus0$
+\end_inset
+
+ de grado
+\begin_inset Formula $n$
+\end_inset
+
+ y
+\begin_inset Formula ${\cal P}\subseteq K[X]\setminus0$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $f$
+\end_inset
+
+ se descompone completamente sobre
+\begin_inset Formula $L$
+\end_inset
+
+ si y solo si lo hace el polinomio mónico
+\begin_inset Formula $f/f_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si
+\begin_inset Formula $n\in\{0,1\}$
+\end_inset
+
+,
+\begin_inset Formula $K$
+\end_inset
+
+ es cuerpo de descomposición de
+\begin_inset Formula $f$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Sean
+\begin_inset Formula $f_{1},\dots,f_{n}\in K[X]\setminus0$
+\end_inset
+
+ y
+\begin_inset Formula $f:=f_{1}\cdots f_{n}$
+\end_inset
+
+,
+\begin_inset Formula $L$
+\end_inset
+
+ es un cuerpo de descomposición de
+\begin_inset Formula $\{f_{1},\dots,f_{n}\}$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+ si y solo si lo es de
+\begin_inset Formula $f$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si
+\begin_inset Formula $L$
+\end_inset
+
+ es un cuerpo de descomposición de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+,
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ es algebraica.
+\end_layout
+
+\begin_layout Enumerate
+Si
+\begin_inset Formula $L$
+\end_inset
+
+ es un cuerpo de descomposición de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+, también lo es de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ sobre cualquier cuerpo intermedio entre
+\begin_inset Formula $K$
+\end_inset
+
+ y
+\begin_inset Formula $L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si cada
+\begin_inset Formula $f\in{\cal P}$
+\end_inset
+
+ se descompone completamente en
+\begin_inset Formula $L$
+\end_inset
+
+ , sea
+\begin_inset Formula $S\subseteq L$
+\end_inset
+
+ el conjunto de raíces de los elementos de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+, entonces
+\begin_inset Formula $K(S)$
+\end_inset
+
+ es un cuerpo de descomposición de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, para obtener el cuerpo de descomposición de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+, basta considerar los polinomios mónicos correspondientes a los polinomios
+ de
+\begin_inset Formula ${\cal P}$
+\end_inset
+
+ de grado al menos 2, u opcionalmente del producto de todos ellos si hay
+ un número finito de ellos, luego hay que encontrar una extensión de
+\begin_inset Formula $K$
+\end_inset
+
+ en que estos polinomios tengan todas sus raíces y quedarnos con el subcuerpo
+ generado por las raíces sobre
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un cuerpo de descomposición de
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ sobre
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es
+\begin_inset Formula $\mathbb{Q}(\xi)$
+\end_inset
+
+, con
+\begin_inset Formula $\xi:=e^{2\pi i/n}$
+\end_inset
+
+, que también es el cuerpo de descomposición de
+\begin_inset Formula $X^{n-1}+\dots+X+1$
+\end_inset
+
+ y de
+\begin_inset Formula $\text{Irr}(\xi,\mathbb{Q})$
+\end_inset
+
+.
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+En efecto, las raíces de
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ en
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ son
+\begin_inset Formula $\{\xi^{i}\}_{i=0}^{n-1}$
+\end_inset
+
+ y
+\begin_inset Formula $\mathbb{Q}(\{\xi^{i}\}_{i=0}^{n-1})=\mathbb{Q}(\xi)$
+\end_inset
+
+.
+ Los otros dos polinomios son divisores de
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ y por tanto tendrán un subconjunto de sus raíces, pero una de las que tienen
+ es
+\begin_inset Formula $\xi$
+\end_inset
+
+ y
+\begin_inset Formula $\mathbb{Q}(\xi)$
+\end_inset
+
+ contiene al resto.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo y
+\begin_inset Formula $f\in K[X]\setminus0$
+\end_inset
+
+, existe un cuerpo de descomposición
+\begin_inset Formula $L$
+\end_inset
+
+ de
+\begin_inset Formula $f$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+ y
+\begin_inset Formula $[L:K]\leq n!$
+\end_inset
+
+.
+ Esta cota no es mejorable
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+; por ejemplo, las raíces de
+\begin_inset Formula $X^{3}-2$
+\end_inset
+
+ en
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ son
+\begin_inset Formula $\alpha$
+\end_inset
+
+,
+\begin_inset Formula $\alpha\omega$
+\end_inset
+
+ y
+\begin_inset Formula $\alpha\omega^{2}$
+\end_inset
+
+ con
+\begin_inset Formula $\alpha:=\sqrt[3]{2}$
+\end_inset
+
+ y
+\begin_inset Formula $\omega:=e^{2\pi i/3}$
+\end_inset
+
+, luego un cuerpo de descomposición es
+\begin_inset Formula $\mathbb{Q}(\alpha,\alpha\omega,\alpha\omega^{2})$
+\end_inset
+
+, y como
+\begin_inset Formula $\omega=\frac{1}{2}\alpha^{2}(\alpha\omega)$
+\end_inset
+
+, esto es lo mismo que
+\begin_inset Formula $\mathbb{Q}(\alpha,\omega)$
+\end_inset
+
+, pero como
+\begin_inset Formula $\text{Irr}(\alpha,\mathbb{Q})=X^{3}-2$
+\end_inset
+
+ e
+\begin_inset Formula $\text{Irr}(\omega,\mathbb{Q}(\alpha))=X^{2}+X+1$
+\end_inset
+
+,
+\begin_inset Formula $[\mathbb{Q}(\alpha,\omega):\mathbb{Q}]=[\mathbb{Q}(\alpha,\omega):\mathbb{Q}(\alpha)][\mathbb{Q}(\alpha):\mathbb{Q}]=2\cdot3=6$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Grupo de Galois de un polinomio
+\end_layout
+
+\begin_layout Standard
+Dados un cuerpo
+\begin_inset Formula $K$
+\end_inset
+
+ y un
+\begin_inset Formula $f\in K[X]$
+\end_inset
+
+ con cuerpo de descomposición
+\begin_inset Formula $L$
+\end_inset
+
+ sobre
+\begin_inset Formula $K$
+\end_inset
+
+, el
+\series bold
+grupo de Galois
+\series default
+ de
+\begin_inset Formula $f$
+\end_inset
+
+ es
+\begin_inset Formula $G_{f}:=\text{Gal}(L/K)$
+\end_inset
+
+.
+ Sean
+\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}\in L$
+\end_inset
+
+ las raíces distintas de
+\begin_inset Formula $f$
+\end_inset
+
+, cada
+\begin_inset Formula $\sigma\in G_{f}$
+\end_inset
+
+ lleva raíces a raíces y por tanto
+\begin_inset Formula $\sigma|_{\{\alpha_{1},\dots,\alpha_{n}\}}:\{\alpha_{1},\dots,\alpha_{n}\}\to\{\alpha_{1},\dots,\alpha_{n}\}$
+\end_inset
+
+ es inyectiva por serlo
+\begin_inset Formula $\sigma$
+\end_inset
+
+ y por tanto biyectiva.
+ Sea
+\begin_inset Formula $\varphi:G_{f}\to{\cal S}_{n}$
+\end_inset
+
+ dada por
+\begin_inset Formula $\varphi(\sigma)(i)=j\iff\sigma(\alpha_{i})=\alpha_{j}$
+\end_inset
+
+,
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es un homomorfismo inyectivo y por tanto
+\begin_inset Formula $G_{f}\cong\text{Im}\varphi\leq{\cal S}_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Clausura algebraica
+\end_layout
+
+\begin_layout Standard
+Un cuerpo
+\begin_inset Formula $K$
+\end_inset
+
+ es
+\series bold
+algebraicamente cerrado
+\series default
+ si todo
+\begin_inset Formula $f\in K[X]\setminus0$
+\end_inset
+
+ tiene una raíz en
+\begin_inset Formula $K$
+\end_inset
+
+.
+ Una extensión
+\begin_inset Formula $L$
+\end_inset
+
+ de un cuerpo
+\begin_inset Formula $K$
+\end_inset
+
+ es una
+\series bold
+clausura algebraica
+\series default
+ de
+\begin_inset Formula $K$
+\end_inset
+
+ si
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ es algebraica y
+\begin_inset Formula $L$
+\end_inset
+
+ es algebraicamente cerrado.
+ Todo cuerpo tiene una clausura algebraica.
+\end_layout
+
+\begin_layout Standard
+El cuerpo de descomposición sobre un cuerpo
+\begin_inset Formula $K$
+\end_inset
+
+ de un
+\begin_inset Formula ${\cal P}\subseteq K[X]$
+\end_inset
+
+ es único salvo isomorfismos.
+\end_layout
+
+\begin_layout Section
+Cuerpos finitos
+\end_layout
+
+\begin_layout Standard
+Dado un anillo
+\begin_inset Formula $A$
+\end_inset
+
+ de característica prima
+\begin_inset Formula $p$
+\end_inset
+
+,
+\begin_inset Formula $h:A\to A$
+\end_inset
+
+ dado por
+\begin_inset Formula $h(a):=a^{p}$
+\end_inset
+
+ es un homomorfismo de anillos, el
+\series bold
+homomorfismo de Frobenius
+\series default
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+, pues conserva el 1,
+\begin_inset Formula $h(ab)=(ab)^{p}=a^{p}b^{p}$
+\end_inset
+
+ y
+\begin_inset Formula $h(a+b)=(a+b)^{p}=\sum_{k=0}^{p}\binom{p}{k}a^{p}b^{p-k}=a^{p}+b^{p}$
+\end_inset
+
+, usando que, para
+\begin_inset Formula $k\in\{1,\dots,p-1\}$
+\end_inset
+
+,
+\begin_inset Formula $\binom{p}{k}=\frac{p!}{(p-k)!k!}=0$
+\end_inset
+
+ al ser un cociente de un múltiplo de
+\begin_inset Formula $p$
+\end_inset
+
+ entre algo que no es múltiplo de
+\begin_inset Formula $p$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+.
+ En particular
+\begin_inset Formula $h^{n}=(a\mapsto a^{p^{n}})$
+\end_inset
+
+ es un homomorfismo.
+\end_layout
+
+\begin_layout Standard
+Como
+\series bold
+teorema
+\series default
+:
+\end_layout
+
+\begin_layout Enumerate
+Si
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo finito, existen
+\begin_inset Formula $p,n\in\mathbb{Z}^{+}$
+\end_inset
+
+ con
+\begin_inset Formula $p$
+\end_inset
+
+ primo tales que
+\begin_inset Formula $\text{car}K=p$
+\end_inset
+
+,
+\begin_inset Formula $|K|=p^{n}$
+\end_inset
+
+ y
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo de descomposición sobre
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ de
+\begin_inset Formula $X^{p^{n}}-X$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Como
+\begin_inset Formula $K$
+\end_inset
+
+ es finito,
+\begin_inset Formula $\text{car}K\neq0$
+\end_inset
+
+ y
+\begin_inset Formula $\text{car}K=p$
+\end_inset
+
+ para cierto primo
+\begin_inset Formula $p$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula $K$
+\end_inset
+
+ es una extensión finita de su subcuerpo primo
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ y, tomando
+\begin_inset Formula $n:=[K:\mathbb{Z}_{p}]$
+\end_inset
+
+,
+\begin_inset Formula $|K|=p^{n}$
+\end_inset
+
+, luego
+\begin_inset Formula $n<\infty$
+\end_inset
+
+.
+ Entonces el grupo multiplicativo
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ tiene
+\begin_inset Formula $p^{n}-1$
+\end_inset
+
+ elementos, pero para
+\begin_inset Formula $g\in K^{*}$
+\end_inset
+
+,
+\begin_inset Formula $g^{|K^{*}|}=g^{p^{n}-1}=1$
+\end_inset
+
+ y
+\begin_inset Formula $g$
+\end_inset
+
+ es raíz de
+\begin_inset Formula $X^{p^{n}-1}-1$
+\end_inset
+
+ y por tanto de
+\begin_inset Formula $f:=X^{p^{n}}-X$
+\end_inset
+
+, que también es raíz del 0.
+ Como
+\begin_inset Formula $f$
+\end_inset
+
+ tiene a lo sumo
+\begin_inset Formula $p^{n}=|K|$
+\end_inset
+
+ raíces,
+\begin_inset Formula $K$
+\end_inset
+
+ está formado por las raíces de
+\begin_inset Formula $f$
+\end_inset
+
+ y por tanto es está generado por estas.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para cada
+\begin_inset Formula $p,n\in\mathbb{Z}^{+}$
+\end_inset
+
+ con
+\begin_inset Formula $p$
+\end_inset
+
+ primo, sea
+\begin_inset Formula $\overline{\mathbb{Z}_{p}}$
+\end_inset
+
+ la clausura algebraica de
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+, el cuerpo de descomposición sobre
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ de
+\begin_inset Formula $X^{p^{n}}-X$
+\end_inset
+
+ tiene
+\begin_inset Formula $p^{n}$
+\end_inset
+
+ elementos y viene dado por
+\begin_inset Formula $\mathbb{F}_{p^{n}}:=\{\alpha\in\overline{\mathbb{Z}_{p}}:\alpha^{p^{n}}=\alpha\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sea
+\begin_inset Formula $S:=\{\alpha\in\overline{\mathbb{Z}_{p}}:\alpha^{p^{n}}=\alpha\}$
+\end_inset
+
+ el conjunto de raíces de
+\begin_inset Formula $f:=X^{p^{n}}-X$
+\end_inset
+
+ en
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+, el cuerpo de descomposición es
+\begin_inset Formula $\mathbb{F}_{p^{n}}=\mathbb{Z}_{p}(S)$
+\end_inset
+
+, pero
+\begin_inset Formula $S$
+\end_inset
+
+ un cuerpo, pues
+\begin_inset Formula $1\in S$
+\end_inset
+
+ y, para
+\begin_inset Formula $\alpha,\beta\in S$
+\end_inset
+
+, por el homomorfismo de Frobenius,
+\begin_inset Formula $(\alpha-\beta)^{p^{n}}=\alpha^{p^{n}}-\beta^{p^{n}}=\alpha-\beta$
+\end_inset
+
+,
+\begin_inset Formula $(\alpha\beta)^{p^{n}}=\alpha^{p^{n}}\beta^{p^{n}}=\alpha\beta$
+\end_inset
+
+ y
+\begin_inset Formula $(\alpha^{-1})^{p^{n}}=(\alpha^{p^{n}})^{-1}=\alpha^{-1}$
+\end_inset
+
+.
+ Además,
+\begin_inset Formula $(1)=\mathbb{Z}_{p}\subseteq S$
+\end_inset
+
+ y
+\begin_inset Formula $\mathbb{Z}_{p}(S)=S$
+\end_inset
+
+, pero como
+\begin_inset Formula $\text{mcd}\{f,f'\}=\text{mcd}\{X^{p^{n}}-X,-1\}=1$
+\end_inset
+
+,
+\begin_inset Formula $f$
+\end_inset
+
+ no tiene raíces múltiples y
+\begin_inset Formula $|\mathbb{F}_{p^{n}}|=|S|=p^{n}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para
+\begin_inset Formula $p,n\in\mathbb{Z}^{+}$
+\end_inset
+
+ con
+\begin_inset Formula $p$
+\end_inset
+
+ primo, todo cuerpo finito de tamaño
+\begin_inset Formula $p^{n}$
+\end_inset
+
+ es isomorfo a
+\begin_inset Formula $\mathbb{F}_{p^{n}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage pagebreak
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Con esto:
+\end_layout
+
+\begin_layout Enumerate
+Vistos como subcuerpos de
+\begin_inset Formula $\overline{\mathbb{Z}_{p}}$
+\end_inset
+
+,
+\begin_inset Formula $\mathbb{F}_{p^{m}}\subseteq\mathbb{F}_{p^{n}}\iff m\mid n$
+\end_inset
+
+.
+\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
+
+
+\begin_inset Formula $\mathbb{Z}_{p}\subseteq\mathbb{F}_{p^{m}}\subseteq\mathbb{F}_{p^{n}}$
+\end_inset
+
+, luego
+\begin_inset Formula $m=[\mathbb{F}_{p^{m}}:\mathbb{Z}_{p}]\mid[\mathbb{F}_{p^{n}}:\mathbb{Z}_{p}]=n$
+\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
+
+Sea
+\begin_inset Formula $t:=\frac{n}{m}$
+\end_inset
+
+, para
+\begin_inset Formula $\alpha$
+\end_inset
+
+ con
+\begin_inset Formula $\alpha=\alpha^{p^{m}}$
+\end_inset
+
+ es
+\begin_inset Formula $\alpha=\alpha^{p^{mt}}=\alpha^{p^{n}}$
+\end_inset
+
+.
+ En efecto, para
+\begin_inset Formula $t=1$
+\end_inset
+
+ esto es trivial, y para
+\begin_inset Formula $t>1$
+\end_inset
+
+, supuesto esto probado para
+\begin_inset Formula $t-1$
+\end_inset
+
+,
+\begin_inset Formula $\alpha^{p^{mt}}=\alpha^{p^{m(t-1)+m}}=\alpha^{p^{m(t-1)}p^{m}}=(\alpha^{p^{m(t-1)}})^{p^{m}}=\alpha^{p^{m}}=\alpha$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $X^{p^{n}}-X$
+\end_inset
+
+ es el producto de todos los irreducibles mónicos de
+\begin_inset Formula $\mathbb{Z}_{p}[X]$
+\end_inset
+
+ cuyo grado divide a
+\begin_inset Formula $n$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Como
+\begin_inset Formula $F:=X^{p^{n}}-X$
+\end_inset
+
+ no tiene raíces múltiples, no tiene factores repetidos y es pues el producto
+ de todos los irreducibles mónicos que lo dividen, y queremos ver que, para
+ un irreducible mónico
+\begin_inset Formula $f\in\mathbb{Z}_{p}[X]$
+\end_inset
+
+ de grado
+\begin_inset Formula $m$
+\end_inset
+
+,
+\begin_inset Formula $f\mid F$
+\end_inset
+
+ si y solo si
+\begin_inset Formula $m\mid n$
+\end_inset
+
+.
+ Sea entonces una raíz
+\begin_inset Formula $\alpha\in\overline{\mathbb{Z}_{p}}$
+\end_inset
+
+ de
+\begin_inset Formula $f$
+\end_inset
+
+,
+\begin_inset Formula $f=\text{Irr}(\alpha,\mathbb{Z}_{p})$
+\end_inset
+
+ y
+\begin_inset Formula $[\mathbb{Z}_{p}(\alpha):\mathbb{Z}_{p}]=m$
+\end_inset
+
+, luego
+\begin_inset Formula $|\mathbb{Z}_{p}(\alpha)|=p^{m}$
+\end_inset
+
+ y por tanto
+\begin_inset Formula $\mathbb{Z}_{p}(\alpha)\cong\mathbb{F}_{p^{m}}$
+\end_inset
+
+.
+ Entonces, por la caracterización de
+\begin_inset Formula $\text{Irr}(\alpha,f)$
+\end_inset
+
+,
+\begin_inset Formula $f\mid F$
+\end_inset
+
+ si y solo si
+\begin_inset Formula $\alpha$
+\end_inset
+
+ es raíz de
+\begin_inset Formula $F$
+\end_inset
+
+, si y solo si
+\begin_inset Formula $\alpha^{p^{n}}=\alpha$
+\end_inset
+
+, si y solo si
+\begin_inset Formula $\alpha\in\mathbb{F}_{p^{n}}$
+\end_inset
+
+, si y solo si
+\begin_inset Formula $\mathbb{F}_{p^{m}}\cong\mathbb{Z}_{p}(\alpha)\subseteq\mathbb{F}_{p^{n}}$
+\end_inset
+
+, si y solo si
+\begin_inset Formula $m\mid n$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Dada una extensión de cuerpos finitos
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ de grado
+\begin_inset Formula $m$
+\end_inset
+
+, existe
+\begin_inset Formula $\alpha\in L$
+\end_inset
+
+ tal que
+\begin_inset Formula $L=K(\alpha)$
+\end_inset
+
+,
+\begin_inset Formula $\text{Irr}(\alpha,K)$
+\end_inset
+
+ tiene
+\begin_inset Formula $m$
+\end_inset
+
+ raíces distintas en
+\begin_inset Formula $L$
+\end_inset
+
+ y
+\begin_inset Formula $|\text{Gal}(L/K)|=m$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-31 01:09:27 +0000