From b01bada353e42059d8c17caac88decee78860410 Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Tue, 8 Jun 2021 19:40:00 +0200
Subject: Algebraicas tema 5

---
 ealg/n.lyx  |   14 +
 ealg/n5.lyx | 1871 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 1885 insertions(+)
 create mode 100644 ealg/n5.lyx

(limited to 'ealg')

diff --git a/ealg/n.lyx b/ealg/n.lyx
index 099164a..b013914 100644
--- a/ealg/n.lyx
+++ b/ealg/n.lyx
@@ -188,6 +188,20 @@ filename "n4.lyx"
 \end_inset
 
 
+\end_layout
+
+\begin_layout Chapter
+Raíces de la unidad
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n5.lyx"
+
+\end_inset
+
+
 \end_layout
 
 \end_body
diff --git a/ealg/n5.lyx b/ealg/n5.lyx
new file mode 100644
index 0000000..6d812c2
--- /dev/null
+++ b/ealg/n5.lyx
@@ -0,0 +1,1871 @@
+#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 $n\geq2$
+\end_inset
+
+, un 
+\begin_inset Formula $\xi\in K$
+\end_inset
+
+ es una 
+\series bold
+raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima de la unidad
+\series default
+ o 
+\series bold
+de uno
+\series default
+ si 
+\begin_inset Formula $\xi^{n}=1$
+\end_inset
+
+, y llamamos
+\begin_inset Formula 
+\[
+{\cal U}_{n}(K):=\{\xi\in K:\xi^{n}=1\}=\{\xi\in K:o_{K^{*}}(\xi)\mid n\}.
+\]
+
+\end_inset
+
+En efecto, el orden de 
+\begin_inset Formula $\xi$
+\end_inset
+
+ en 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ es el menor 
+\begin_inset Formula $m>0$
+\end_inset
+
+ con 
+\begin_inset Formula $\xi^{m}=n$
+\end_inset
+
+, luego si 
+\begin_inset Formula $m\mid n$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\xi^{n}=(\xi^{m})^{n/m}=1^{n/m}=1$
+\end_inset
+
+ y si 
+\begin_inset Formula $\xi^{n}=1$
+\end_inset
+
+, sean 
+\begin_inset Formula $q$
+\end_inset
+
+ y 
+\begin_inset Formula $r$
+\end_inset
+
+ el cociente y resto de 
+\begin_inset Formula $n/m$
+\end_inset
+
+, entonces 
+\begin_inset Formula $1=\xi^{mq+r}=(\xi^{m})^{q}\xi^{r}=\xi^{r}$
+\end_inset
+
+, pero como 
+\begin_inset Formula $r<m$
+\end_inset
+
+ debe ser 
+\begin_inset Formula $r=0$
+\end_inset
+
+ y 
+\begin_inset Formula $mq=n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula ${\cal U}_{n}(K)$
+\end_inset
+
+ es un subgrupo cíclico de 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+, pues contiene al 1, es cerrado por productos (
+\begin_inset Formula $\xi^{n}=1\land\mu^{n}=1\implies(\xi\mu)^{n}=\xi^{n}\mu^{n}=1$
+\end_inset
+
+) y, como 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ es cíclico, todos sus subgrupos también.
+ Una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima es 
+\series bold
+primitiva
+\series default
+ si 
+\begin_inset Formula $o_{K^{*}}(\xi)=n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Propiedades: Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es un cuerpo y 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Toda raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima de uno es raíz 
+\begin_inset Formula $tn$
+\end_inset
+
+-ésima de uno para 
+\begin_inset Formula $t\geq1$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\xi^{n}=1\implies\xi^{tn}=(\xi^{n})^{t}=1^{t}=1$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+1 no es raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $o(1)=1$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{car}K\neq2$
+\end_inset
+
+, 
+\begin_inset Formula $-1$
+\end_inset
+
+ es raíz cuadrada primitiva de uno.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $-1\neq1,(-1)^{2}=1$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\xi\in K$
+\end_inset
+
+ es raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno si y sólo si 
+\begin_inset Formula $|{\cal U}_{n}(K)|=n$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal U}_{n}(K)=\langle\xi\rangle$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\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 $|{\cal U}_{n}(K)|\geq|\langle\xi\rangle|=n$
+\end_inset
+
+, pero como 
+\begin_inset Formula ${\cal U}_{n}(K)$
+\end_inset
+
+ lo forman las raíces de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+, 
+\begin_inset Formula $|{\cal U}_{n}(K)|\leq n$
+\end_inset
+
+, luego 
+\begin_inset Formula $|{\cal U}_{n}(K)|=n$
+\end_inset
+
+ y 
+\begin_inset Formula $\langle\xi\rangle={\cal U}_{n}(K)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\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 $o(\xi)=|\langle\xi\rangle|=|{\cal U}_{n}(K)|=n$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $K$
+\end_inset
+
+ contiene alguna raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno si y sólo si 
+\begin_inset Formula $|{\cal U}_{n}(K)|=n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es finito, contiene alguna raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno si y sólo si 
+\begin_inset Formula $n\mid|K|-1$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Como 
+\begin_inset Formula $K^{*}$
+\end_inset
+
+ es cíclico, tiene elementos de todos los órdenes que dividen a 
+\begin_inset Formula $|K^{*}|=|K|-1$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal U}_{n}(\mathbb{C})=\langle e^{2\pi i/n}\rangle$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula 
+\[
+{\cal U}_{n}(\mathbb{R})=\begin{cases}
+\{\pm1\}, & n\text{ es par};\\
+\{1\}, & n\text{ es impar}.
+\end{cases}
+\]
+
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula ${\cal U}_{n}(\mathbb{R})\subseteq{\cal U}_{n}(\mathbb{C})$
+\end_inset
+
+, y si 
+\begin_inset Formula $k\in\{0,\dots,n-1\}$
+\end_inset
+
+, 
+\begin_inset Formula $e^{2k\pi i/n}\in\mathbb{R}\iff2k\pi/n\in\pi\mathbb{Z}\iff2k\in n\mathbb{Z}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $n$
+\end_inset
+
+ es par, esto lo cumplen 
+\begin_inset Formula $k=0$
+\end_inset
+
+ y 
+\begin_inset Formula $k=n/2$
+\end_inset
+
+, y si 
+\begin_inset Formula $n$
+\end_inset
+
+ es impar, solo lo cumple 
+\begin_inset Formula $k=0$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Ni 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ ni ningún subcuerpo suyo contienen raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas primitivas para 
+\begin_inset Formula $n\geq3$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Estas tendrían orden al menos 3, pero 
+\begin_inset Formula $o(1)=1$
+\end_inset
+
+ y 
+\begin_inset Formula $o(-1)=2$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{F}_{4}$
+\end_inset
+
+ tiene 2 raíces cúbicas primitivas; 
+\begin_inset Formula $\mathbb{F}_{8}$
+\end_inset
+
+ tiene 6 raíces séptimas primitivas, y 
+\begin_inset Formula $\mathbb{F}_{9}$
+\end_inset
+
+ tiene 4 raíces octavas primitivas y 2 raíces cuartas primitivas.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\text{car}K=p\neq0$
+\end_inset
+
+, la única raíz 
+\begin_inset Formula $p$
+\end_inset
+
+-ésima es 1.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Toda raíz 
+\begin_inset Formula $p$
+\end_inset
+
+-ésima, es raíz de 
+\begin_inset Formula $X^{p}-1=(X-1)^{p}$
+\end_inset
+
+ por el homomorfismo de Frobenius, luego debe ser 1.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+ y 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+, existe una extensión 
+\begin_inset Formula $L$
+\end_inset
+
+ de 
+\begin_inset Formula $K$
+\end_inset
+
+ que contiene raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas primitivas de la unidad si y sólo si 
+\begin_inset Formula $\text{car}K\nmid n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\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 $L$
+\end_inset
+
+ un cuerpo de descomposición sobre 
+\begin_inset Formula $K$
+\end_inset
+
+ de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+, como 
+\begin_inset Formula $f'=nX^{n-1}$
+\end_inset
+
+ y 
+\begin_inset Formula $n\neq0$
+\end_inset
+
+ al ser 
+\begin_inset Formula $\text{car}K\nmid n$
+\end_inset
+
+, la única raíz de 
+\begin_inset Formula $f'$
+\end_inset
+
+ es 0 y por tanto 
+\begin_inset Formula $f$
+\end_inset
+
+ no tiene raíces múltiples, luego tiene 
+\begin_inset Formula $n$
+\end_inset
+
+ raíces raíces distintas y 
+\begin_inset Formula $|{\cal U}_{n}(L)|=n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Probamos el contrarrecíproco.
+ Si 
+\begin_inset Formula $p:=\text{car}K\mid n$
+\end_inset
+
+, existe 
+\begin_inset Formula $t\in\mathbb{N}$
+\end_inset
+
+ con 
+\begin_inset Formula $n=tp$
+\end_inset
+
+ y 
+\begin_inset Formula $X^{n}-1=X^{tp}-1^{p}=(X^{t}-1)^{p}$
+\end_inset
+
+ por el homomorfismo de Frobenius, luego 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ tiene a lo sumo 
+\begin_inset Formula $t=n/p<n$
+\end_inset
+
+ raíces y por tanto no tiene raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas de uno primitivas.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{GyA}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si [
+\begin_inset Formula $G$
+\end_inset
+
+ es un grupo,] 
+\begin_inset Formula $a$
+\end_inset
+
+[
+\begin_inset Formula $\in G$
+\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
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{CyN}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Definimos la 
+\series bold
+función 
+\begin_inset Formula $\phi$
+\end_inset
+
+ de Euler
+\series default
+ como 
+\begin_inset Formula $\phi:\mathbb{N}\rightarrow\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\phi(m)=|\{x\in\mathbb{N}|1\leq x\leq m\land\text{mcd}(x,m)=1\}|=|\mathbb{Z}_{m}^{*}|$
+\end_inset
+
+.
+ [...] Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\phi(p^{n})=p^{n-1}(p-1)$
+\end_inset
+
+.
+ [...] Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\phi(p^{n})=p^{n-1}(p-1)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+ tiene una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno 
+\begin_inset Formula $\xi$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $K$
+\end_inset
+
+ tiene exactamente 
+\begin_inset Formula $n$
+\end_inset
+
+ raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas de uno, 
+\begin_inset Formula $\xi,\xi^{2},\dots,\xi^{n}=1$
+\end_inset
+
+, y 
+\begin_inset Formula $\phi(n)$
+\end_inset
+
+ de ellas son primitivas.
+ En particular 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ se descompone completamente en 
+\begin_inset Formula $K[X]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para cada 
+\begin_inset Formula $d\mid n$
+\end_inset
+
+ natural hay una raíz 
+\begin_inset Formula $d$
+\end_inset
+
+-ésima primitiva en 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula $\xi^{n/d}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $K$
+\end_inset
+
+ es finito, esto se cumple para 
+\begin_inset Formula $n=|K|-1$
+\end_inset
+
+, y si 
+\begin_inset Formula $K\subseteq\mathbb{C}$
+\end_inset
+
+, se aplica cuando 
+\begin_inset Formula $e^{2\pi i/n}\in K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Polinomios ciclotómicos
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $P$
+\end_inset
+
+ un cuerpo primo (
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+), 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{car}P\nmid n$
+\end_inset
+
+ y 
+\begin_inset Formula $L$
+\end_inset
+
+ el cuerpo de descomposición sobre 
+\begin_inset Formula $P$
+\end_inset
+
+ de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+, que contiene 
+\begin_inset Formula $\phi(n)$
+\end_inset
+
+ raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas primitivas de uno 
+\begin_inset Formula $\xi_{1},\dots,\xi_{n}$
+\end_inset
+
+, llamamos 
+\series bold
+
+\begin_inset Formula $n$
+\end_inset
+
+-ésimo polinomio ciclotómico en característica 
+\begin_inset Formula $\text{car}P$
+\end_inset
+
+
+\series default
+ a 
+\begin_inset Formula 
+\[
+\Phi_{n}(X):=(X-\xi_{1})\cdots(X-\xi_{r})\in L[X].
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $\text{car}K\nmid n$
+\end_inset
+
+, 
+\begin_inset Formula $X^{n}-1=\prod_{0<d\mid n}\Phi_{d}(X)$
+\end_inset
+
+, con el convenio de que 
+\begin_inset Formula $\Phi_{1}(X)=X-1$
+\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
+Si 
+\begin_inset Formula $P$
+\end_inset
+
+ es un cuerpo primo:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $q\neq\text{car}P$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\Phi_{q}(X)=X^{q-1}+\dots+X+1$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $X^{q}-1=\prod_{d\mid q}\Phi_{d}(X)=(X-1)\Phi_{q}(X)$
+\end_inset
+
+, y 
+\begin_inset Formula $\frac{X^{q}-1}{X-1}=X^{q-1}+\dots+X+1$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $n\geq3$
+\end_inset
+
+ es impar, 
+\begin_inset Formula $\Phi_{2n}(X)=\Phi_{n}(-X)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Como 
+\begin_inset Formula $\text{mcd}(n,2)=1$
+\end_inset
+
+, 
+\begin_inset Formula $\phi(2n)=\phi(2)\phi(n)=\phi(n)$
+\end_inset
+
+, luego 
+\begin_inset Formula $\text{gr}\Phi_{2n}=\text{gr}\Phi_{n}$
+\end_inset
+
+ y, como ninguno tiene raíces múltiples, basta ver que 
+\begin_inset Formula $\Phi_{2n}(X)$
+\end_inset
+
+ tiene las raíces de 
+\begin_inset Formula $\Phi_{n}(-X)$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $G$
+\end_inset
+
+ es un grupo abeliano, 
+\begin_inset Formula $x,y\in G$
+\end_inset
+
+ y 
+\begin_inset Formula $m:=o(x)$
+\end_inset
+
+ y 
+\begin_inset Formula $n:=o(y)$
+\end_inset
+
+ son coprimos, entonces 
+\begin_inset Formula $o(xy)=mn$
+\end_inset
+
+, pues 
+\begin_inset Formula $(xy)^{mn}=(x^{m})^{n}(y^{n})^{m}=1^{n}1^{m}=1$
+\end_inset
+
+ y, si 
+\begin_inset Formula $k\in\mathbb{N}^{*}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $(xy)^{k}=1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $x^{k}=y^{-k}\in\langle x\rangle\cap\langle y\rangle$
+\end_inset
+
+ y por el teorema de Lagrange es 
+\begin_inset Formula $|\langle x\rangle\cap\langle y\rangle|\mid m,n$
+\end_inset
+
+, luego 
+\begin_inset Formula $|\langle x\rangle\cap\langle y\rangle|=1$
+\end_inset
+
+, 
+\begin_inset Formula $x^{k}=y^{-k}=1$
+\end_inset
+
+, 
+\begin_inset Formula $m,n\mid k$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{mcm}\{m,n\}=mn\mid k$
+\end_inset
+
+.
+ En nuestro caso, 
+\begin_inset Formula $o(-\xi)=n$
+\end_inset
+
+, pero 
+\begin_inset Formula $o(\xi)=o((-1)(-\xi))=o(-1)o(-\xi)$
+\end_inset
+
+.
+ Como no estamos en característica 2 ya que de estarlo 
+\begin_inset Formula $\Phi_{2n}$
+\end_inset
+
+ no estaría definido, 
+\begin_inset Formula $o(-1)=2$
+\end_inset
+
+, luego 
+\begin_inset Formula $o(\xi)=2o(-\xi)=2n$
+\end_inset
+
+ y 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{2n}(X)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo y 
+\begin_inset Formula $k\geq1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\Phi_{p^{k}}(X)=\Phi_{p}(X^{p^{k-1}})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\text{gr\ensuremath{\Phi_{p_{k}}=}}\phi(p^{k})=(p-1)p^{k-1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}\Phi_{p}(X^{p^{k-1}})=\phi(p)p^{k-1}=(p-1)p^{k-1}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{p_{k}}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $o(\xi)=p^{k}$
+\end_inset
+
+ y 
+\begin_inset Formula $o(\xi^{p^{k-1}})=p$
+\end_inset
+
+, luego 
+\begin_inset Formula $\xi^{p^{k-1}}$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{p}$
+\end_inset
+
+ y 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi^{p^{k-1}}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $n=p_{1}^{r_{1}}\cdots p_{s}^{r_{s}}$
+\end_inset
+
+ con los 
+\begin_inset Formula $p_{i}$
+\end_inset
+
+ primos distintos, 
+\begin_inset Formula $\Phi_{n}(X)=\Phi_{p_{1}\cdots p_{s}}(X^{p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $p$
+\end_inset
+
+ es primo y no divide a 
+\begin_inset Formula $n$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\Phi_{pn}(X)\Phi_{n}(X)=\Phi_{n}(X^{p})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Phi_{n}\in P[X]$
+\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
+En general los polinomios ciclotómicos no son irreducibles, pues por ejemplo
+ en 
+\begin_inset Formula $\mathbb{Z}_{7}$
+\end_inset
+
+ las raíces terceras primitivas son 2 y 4 y 
+\begin_inset Formula $\Phi_{3}(X)=(X-2)(X-4)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Extensiones ciclotómicas
+\end_layout
+
+\begin_layout Standard
+Una extensión 
+\begin_inset Formula $K\subseteq F$
+\end_inset
+
+ es una 
+\series bold
+ciclotómica de orden 
+\begin_inset Formula $n$
+\end_inset
+
+
+\series default
+ o 
+\begin_inset Formula $F$
+\end_inset
+
+ es el 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimo 
+\series bold
+cuerpo ciclotómico
+\series default
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+ si 
+\begin_inset Formula $F$
+\end_inset
+
+ es el cuerpo de descomposición de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+, y 
+\begin_inset Formula $F$
+\end_inset
+
+ también es el cuerpo ciclotómico sobre cualquier 
+\begin_inset Formula $K'$
+\end_inset
+
+ entre 
+\begin_inset Formula $K$
+\end_inset
+
+ y 
+\begin_inset Formula $F$
+\end_inset
+
+.
+ Cada cuerpo tiene una extensión ciclotómica de cada orden, única salvo
+ isomorfismos.
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{Q}\subseteq\mathbb{Q}(e^{2\pi i/n})$
+\end_inset
+
+ es una extensión ciclotómica de orden 
+\begin_inset Formula $n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{Z}_{p}\subseteq\mathbb{F}_{p^{n}}$
+\end_inset
+
+ es ciclotómica de orden 
+\begin_inset Formula $p^{n}-1$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Los elementos no nulos de 
+\begin_inset Formula $\mathbb{F}_{p^{n}}$
+\end_inset
+
+ son las raíces de 
+\begin_inset Formula $X^{p^{n}-1}-1$
+\end_inset
+
+ sobre 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+La extensión ciclotómica de orden 
+\begin_inset Formula $n$
+\end_inset
+
+ sobre 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ con 
+\begin_inset Formula $p\nmid n$
+\end_inset
+
+ es 
+\begin_inset Formula $\mathbb{F}_{p^{m}}$
+\end_inset
+
+, siendo 
+\begin_inset Formula $m$
+\end_inset
+
+ el orden de 
+\begin_inset Formula $p$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{Z}_{n}^{*}$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Las extensiones finitas de 
+\begin_inset Formula $\mathbb{Z}_{p}$
+\end_inset
+
+ son de la forma 
+\begin_inset Formula $\mathbb{F}_{p^{k}}$
+\end_inset
+
+, y 
+\begin_inset Formula $\mathbb{F}_{p^{k}}$
+\end_inset
+
+ contiene una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno si y sólo si hay elementos de orden 
+\begin_inset Formula $n$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{F}_{p^{m}}^{*}\cong\mathbb{Z}_{p^{m}-1}$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $n\mid p^{m}-1$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $p^{m}\equiv1\bmod n$
+\end_inset
+
+, y el menor 
+\begin_inset Formula $m$
+\end_inset
+
+ con esa propiedad es 
+\begin_inset Formula $o(p)$
+\end_inset
+
+ en 
+\begin_inset Formula $\mathbb{Z}_{n}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dado un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+ con 
+\begin_inset Formula $p:=\text{car}K\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $m\in\mathbb{Z}^{+}$
+\end_inset
+
+ con 
+\begin_inset Formula $p\nmid m$
+\end_inset
+
+, para 
+\begin_inset Formula $r\in\mathbb{N}$
+\end_inset
+
+, las extensiones ciclotómicas de órdenes 
+\begin_inset Formula $m$
+\end_inset
+
+ y 
+\begin_inset Formula $p^{r}m$
+\end_inset
+
+ coinciden.
+ En efecto, por el homomorfismo de Frobenius, 
+\begin_inset Formula $(\xi^{m}-1)^{p^{r}}=\xi^{p^{r}m}-1$
+\end_inset
+
+, luego 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $X^{p^{r}m}-1$
+\end_inset
+
+ si y sólo si lo es de 
+\begin_inset Formula $X^{m}-1$
+\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
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $\text{car}K\nmid n$
+\end_inset
+
+ y 
+\begin_inset Formula $F$
+\end_inset
+
+ es una extensión ciclotómica de orden 
+\begin_inset Formula $n$
+\end_inset
+
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $F=K(\xi)$
+\end_inset
+
+, siendo 
+\begin_inset Formula $\xi$
+\end_inset
+
+ una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Como 
+\begin_inset Formula $\text{car}F=\text{car}K\nmid n$
+\end_inset
+
+, existe una extensión 
+\begin_inset Formula $M$
+\end_inset
+
+ de 
+\begin_inset Formula $F$
+\end_inset
+
+ con una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva 
+\begin_inset Formula $\xi$
+\end_inset
+
+, luego 
+\begin_inset Formula $\xi\in F$
+\end_inset
+
+, 
+\begin_inset Formula $K(\xi)\subseteq F$
+\end_inset
+
+ y, como el resto de raíces de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ son potencias de 
+\begin_inset Formula $\xi$
+\end_inset
+
+, 
+\begin_inset Formula $F\subseteq K(\xi)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\text{Gal}(F/K)$
+\end_inset
+
+ tiene tamaño 
+\begin_inset Formula $[F:K]$
+\end_inset
+
+ y es isomorfo a un subgrupo de 
+\begin_inset Formula $\mathbb{Z}_{n}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $|\text{Gal}(K(\xi)/K)|=[K(\xi):K]$
+\end_inset
+
+ porque el resto de raíces de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ y por tanto de 
+\begin_inset Formula $\text{Irr}(\xi,K)$
+\end_inset
+
+ están en 
+\begin_inset Formula $K(\xi)$
+\end_inset
+
+.
+ Cada 
+\begin_inset Formula $\sigma\in\text{Gal}(F/K)$
+\end_inset
+
+ lleva 
+\begin_inset Formula $\xi$
+\end_inset
+
+ a una raíz 
+\begin_inset Formula $\xi^{j}$
+\end_inset
+
+ de 
+\begin_inset Formula $X^{n}-1$
+\end_inset
+
+ que debe tener el mismo orden que 
+\begin_inset Formula $\xi$
+\end_inset
+
+ por ser 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ inyectiva, luego 
+\begin_inset Formula $\sigma(\xi)=\xi^{j}$
+\end_inset
+
+ para cierto 
+\begin_inset Formula $j$
+\end_inset
+
+ coprimo con 
+\begin_inset Formula $n$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f:\text{Gal}(F/K)\to\mathbb{Z}_{n}^{*}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f(\sigma)=j\iff\sigma(\xi)=\xi^{j}$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ está bien definida, es inyectiva y es un homomorfismo, ya que 
+\begin_inset Formula $f(1)=1$
+\end_inset
+
+ y, llamando 
+\begin_inset Formula $\sigma_{j}\in\text{Gal}(F/K)$
+\end_inset
+
+ al elemento con 
+\begin_inset Formula $\sigma_{j}(\xi)=\xi^{j}$
+\end_inset
+
+, 
+\begin_inset Formula $f(\sigma_{j}\sigma_{k})=f(\sigma_{jk})=jk=f(\sigma_{j})f(\sigma_{k})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $n$
+\end_inset
+
+ es primo, 
+\begin_inset Formula $\text{Gal}(F/K)$
+\end_inset
+
+ es un cíclico.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\mathbb{Z}_{n}^{*}$
+\end_inset
+
+ es cíclico y por tanto sus subgrupos también.
+\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 Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno en 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\Phi_{n}(X)=\text{Irr}(\xi,\mathbb{Q})$
+\end_inset
+
+.
+ Así, si 
+\begin_inset Formula $\xi=e^{2\pi i/n}$
+\end_inset
+
+, 
+\begin_inset Formula $[\mathbb{Q}(\xi):\mathbb{Q}]=\phi(n)$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Gal}(\mathbb{Q}(\xi)/\mathbb{Q})\cong\mathbb{Z}_{n}^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $n,m\in\mathbb{Z}^{+}$
+\end_inset
+
+ son coprimos y, para 
+\begin_inset Formula $r\in\mathbb{Z}^{+}$
+\end_inset
+
+, 
+\begin_inset Formula $\xi_{r}\in\mathbb{C}$
+\end_inset
+
+ es cualquier raíz 
+\begin_inset Formula $r$
+\end_inset
+
+-ésima primitiva de uno, entonces 
+\begin_inset Formula $\mathbb{Q}(\xi_{n})\mathbb{Q}(\xi_{m})=\mathbb{Q}(\xi_{nm})$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{Q}(\xi_{n})\cap\mathbb{Q}(\xi_{m})=\mathbb{Q}$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
-- 
cgit v1.2.3