From 6ec63727b831a49824c0d1705af9db68ac3fb596 Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Mon, 14 Jun 2021 00:39:19 +0200
Subject: Errata + proposiciones de ejercicios

---
 ealg/n5.lyx | 352 +++++++++++++++++++++++++++++++++++++++++++++++++-----------
 ealg/n6.lyx | 202 +++++++++++++++++++++++++++++++++-
 ealg/n7.lyx |  54 +++++++++-
 3 files changed, 546 insertions(+), 62 deletions(-)

(limited to 'ealg')

diff --git a/ealg/n5.lyx b/ealg/n5.lyx
index c3f6450..fcaa366 100644
--- a/ealg/n5.lyx
+++ b/ealg/n5.lyx
@@ -575,6 +575,59 @@ Toda raíz
 \end_inset
 
 
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+vspace{-0.8em}
+\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
+6.
+\end_layout
+
+\end_inset
+
+Una extensión finita de 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ tiene solo un número finito de raíces de uno.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -1043,10 +1096,10 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $X^{q}-1=\prod_{d\mid q}\Phi_{d}(X)=(X-1)\Phi_{q}(X)$
 \end_inset
 
@@ -1057,7 +1110,11 @@ Si
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $n\geq3$
@@ -1068,10 +1125,10 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 Como 
 \begin_inset Formula $\text{mcd}(n,2)=1$
 \end_inset
@@ -1182,7 +1239,11 @@ Como
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $p$
@@ -1197,10 +1258,10 @@ Si
 \end_inset
 
 .
-\end_layout
+\begin_inset Note Comment
+status open
 
-\begin_deeper
-\begin_layout Standard
+\begin_layout Plain Layout
 \begin_inset Formula $\text{gr\ensuremath{\Phi_{p_{k}}=}}\phi(p^{k})=(p-1)p^{k-1}$
 \end_inset
 
@@ -1244,7 +1305,11 @@ Si
 .
 \end_layout
 
-\end_deeper
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $n=p_{1}^{r_{1}}\cdots p_{s}^{r_{s}}$
@@ -1259,6 +1324,93 @@ Si
 \end_inset
 
 .
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Para empezar, 
+\begin_inset Formula $\text{gr}(\Phi_{n}(X))=\phi(n)=\phi(p_{1}^{r_{1}}\cdots p_{s}^{r_{s}})=(p_{1}-1)\cdots(p_{s}-1)p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Phi_{p_{1}\cdots p_{s}}(X^{p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}})=p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}\phi(p_{1}\cdots p_{s})=(p_{1}-1)\cdots(p_{s}-1)p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}$
+\end_inset
+
+.
+ Sea ahora 
+\begin_inset Formula $\xi$
+\end_inset
+
+ una raíz de 
+\begin_inset Formula $\Phi_{n}(X)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de uno, de modo que 
+\begin_inset Formula $\xi^{p_{1}^{r_{1}}\cdots p_{s}^{r_{s}}}=1$
+\end_inset
+
+, pero 
+\begin_inset Formula $\xi^{p_{1}^{r_{1}}\cdots p_{s}^{r_{s}}}=(\xi^{p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}})^{p_{1}\cdots p_{s}}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\xi^{p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}}$
+\end_inset
+
+ es raíz 
+\begin_inset Formula $(p_{1}\cdots p_{s})$
+\end_inset
+
+-ésima primitiva de uno (si fuera una raíz 
+\begin_inset Formula $k$
+\end_inset
+
+-ésima de uno para 
+\begin_inset Formula $k<p_{1}\cdots p_{s}$
+\end_inset
+
+, 
+\begin_inset Formula $\xi$
+\end_inset
+
+ sería una raíz 
+\begin_inset Formula $(p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}k<n)$
+\end_inset
+
+-ésima de uno
+\begin_inset Formula $\#$
+\end_inset
+
+), luego 
+\begin_inset Formula $\xi^{p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}}$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{p_{1}\cdots p_{s}}$
+\end_inset
+
+ y 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{p_{1}\cdots p_{s}}(X^{p_{1}^{r_{1}-1}\cdots p_{s}^{r_{s}-1}})$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Enumerate
@@ -1274,9 +1426,107 @@ Si
 \begin_inset Formula $\Phi_{pn}(X)\Phi_{n}(X)=\Phi_{n}(X^{p})$
 \end_inset
 
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Como 
+\begin_inset Formula $\text{gr}(\Phi_{n}\Phi_{pn})=\text{gr}n+\text{gr}pn=\phi(n)+(p-1)\phi(n)=p\phi(n)$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{gr}(\Phi_{n}(X^{p}))=p\text{gr}\Phi_{n}=p\phi(n)$
+\end_inset
+
+, y ninguno de los dos tiene raíces múltiples, basta ver que toda raíz 
+\begin_inset Formula $\xi$
+\end_inset
+
+ de 
+\begin_inset Formula $\Phi_{n}\Phi_{pn}$
+\end_inset
+
+ lo es de 
+\begin_inset Formula $\Phi_{n}(X^{p})$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{n}$
+\end_inset
+
+, es raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva del uno y por tanto 
+\begin_inset Formula $\xi^{p}$
+\end_inset
+
+ también lo es, ya que 
+\begin_inset Formula $o(\xi^{p})=\frac{o(\xi)}{\text{mcd}\{o(\xi),p\}}=\frac{n}{\text{mcd}\{n,p\}}=n$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\xi^{p}$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{n}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{pn}$
+\end_inset
+
+, es raíz 
+\begin_inset Formula $pn$
+\end_inset
+
+-ésima primitiva de uno, y como 
+\begin_inset Formula $o(\xi^{p})=\frac{o(\xi)}{\text{mcd}\{o(\xi),p\}}=\frac{pn}{\text{mcd}\{pn,p\}}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\xi^{p}$
+\end_inset
+
+ es una raíz 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima primitiva de 1 y es raíz de 
+\begin_inset Formula $\Phi_{n}$
+\end_inset
+
+.
+ En ambos casos 
+\begin_inset Formula $\xi$
+\end_inset
+
+ es raíz de 
+\begin_inset Formula $\Phi_{n}(X^{p})$
+\end_inset
+
 .
 \end_layout
 
+\end_inset
+
+
+\end_layout
+
 \begin_layout Enumerate
 \begin_inset Formula $\Phi_{n}\in P[X]$
 \end_inset
@@ -1300,19 +1550,6 @@ end{samepage}
 
 \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
@@ -1377,10 +1614,7 @@ cuerpo ciclotómico
 .
  Cada cuerpo tiene una extensión ciclotómica de cada orden, única salvo
  isomorfismos.
-\end_layout
-
-\begin_layout Standard
-Ejemplos:
+ Ejemplos:
 \end_layout
 
 \begin_layout Enumerate
@@ -1444,16 +1678,8 @@ La extensión ciclotómica de orden
 \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 $\mathbb{Z}_{n}^{*}$
+, con 
+\begin_inset Formula $m:=o_{\mathbb{Z}_{n}^{*}}(p)$
 \end_inset
 
 .
@@ -1787,23 +2013,17 @@ Si
 
 \end_deeper
 \begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Plain Layout
-
-
-\backslash
-end{samepage}
-\end_layout
-
+En general los polinomios ciclotómicos no son irreducibles en el cuerpo
+ primo, 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 Standard
-Como 
+.
+ Sin embargo, como 
 \series bold
 teorema
 \series default
@@ -1823,9 +2043,8 @@ teorema
 \begin_inset Formula $\Phi_{n}(X)=\text{Irr}(\xi,\mathbb{Q})$
 \end_inset
 
-.
- Así, si 
-\begin_inset Formula $\xi=e^{2\pi i/n}$
+, luego si 
+\begin_inset Formula $\xi:=e^{2\pi i/n}$
 \end_inset
 
 , 
@@ -1837,10 +2056,7 @@ teorema
 \end_inset
 
 .
-\end_layout
-
-\begin_layout Standard
-Si 
+ Si 
 \begin_inset Formula $n,m\in\mathbb{Z}^{+}$
 \end_inset
 
@@ -1867,5 +2083,21 @@ Si
 .
 \end_layout
 
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{samepage}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document
diff --git a/ealg/n6.lyx b/ealg/n6.lyx
index 50e4d95..da24dc8 100644
--- a/ealg/n6.lyx
+++ b/ealg/n6.lyx
@@ -161,6 +161,27 @@ Ejemplos:
  es normal.
 \end_layout
 
+\begin_layout Enumerate
+\begin_inset Formula $K\subseteq K$
+\end_inset
+
+ es normal.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Los irreducibles en 
+\begin_inset Formula $K$
+\end_inset
+
+ con una raíz en 
+\begin_inset Formula $K$
+\end_inset
+
+ son de grado 1.
+\end_layout
+
+\end_deeper
 \begin_layout Enumerate
 Si 
 \begin_inset Formula $[L:K]=2$
@@ -183,7 +204,7 @@ Los irreducibles en
 \begin_inset Formula $\alpha\in L$
 \end_inset
 
- tiene grado 
+ tienen grado 
 \begin_inset Formula $\text{gr}\text{Irr}(\alpha,K)\leq2$
 \end_inset
 
@@ -784,6 +805,103 @@ Que
 \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 $K\subseteq L$
+\end_inset
+
+ es normal, 
+\begin_inset Formula $K\subseteq E$
+\end_inset
+
+ lo es si y sólo si 
+\begin_inset Formula $E$
+\end_inset
+
+ es 
+\series bold
+estable
+\series default
+ en 
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+, es decir, si 
+\begin_inset Formula $\forall\sigma\in\text{Gal}(L/K),\sigma(E)=E$
+\end_inset
+
+.
+ Una extensión 
+\begin_inset Formula $K\subseteq L$
+\end_inset
+
+ es normal si y sólo si existe una extensión 
+\begin_inset Formula $L\subseteq N$
+\end_inset
+
+ con 
+\begin_inset Formula $K\subseteq N$
+\end_inset
+
+ normal y tal que todo 
+\begin_inset Formula $K$
+\end_inset
+
+-encaje de 
+\begin_inset Formula $L$
+\end_inset
+
+ en 
+\begin_inset Formula $N$
+\end_inset
+
+ es un 
+\begin_inset Formula $K$
+\end_inset
+
+-automorfismo de 
+\begin_inset Formula $L$
+\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
 Sean 
 \begin_inset Formula $K\subseteq L$
@@ -1616,6 +1734,88 @@ Si
 \end_layout
 
 \end_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+vspace{-0.8em}{
+\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 $p:=\text{car}K\neq0$
+\end_inset
+
+, 
+\begin_inset Formula $K$
+\end_inset
+
+ es perfecto si y sólo si todo 
+\begin_inset Formula $a\in K$
+\end_inset
+
+ tiene una raíz 
+\begin_inset Formula $p$
+\end_inset
+
+-ésima en 
+\begin_inset Formula $K$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+5.
+\end_layout
+
+\end_inset
+
+Una extensión algebraica de un cuerpo perfecto es perfecta.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 Además:
 \end_layout
diff --git a/ealg/n7.lyx b/ealg/n7.lyx
index 9e463a3..88407e4 100644
--- a/ealg/n7.lyx
+++ b/ealg/n7.lyx
@@ -128,7 +128,7 @@ correspondencia
 \series bold
 conexión de Galois
 \series default
- asicada a 
+ asociada a 
 \begin_inset Formula $K\subseteq L$
 \end_inset
 
@@ -1612,6 +1612,58 @@ Sean
 
 .
  
+\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
+
+ tiene grado 2 y 
+\begin_inset Formula $\text{car}K\neq2$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Gal}(L/K)\cong C_{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+}}
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \end_body
-- 
cgit v1.2.3