From 2db90bfe4b174570424c408f04000902411d8755 Mon Sep 17 00:00:00 2001
From: Joshua Baer <the.baer.joshua@gmail.ch>
Date: Mon, 12 Apr 2021 21:51:55 +0200
Subject: update to current state of book

---
 vorlesungen/slides/4/galois/aufloesbarkeit.tex | 240 ++++++++++++-------------
 1 file changed, 120 insertions(+), 120 deletions(-)

(limited to 'vorlesungen/slides/4/galois/aufloesbarkeit.tex')

diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
index ef5902b..3d52b00 100644
--- a/vorlesungen/slides/4/galois/aufloesbarkeit.tex
+++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
@@ -1,120 +1,120 @@
-%
-% aufloesbarkeit.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Auflösbarkeit}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Radikalerweiterung}
-Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$
-einer Radikalerweiterung
-\[
-\Bbbk \subset \Bbbk(\alpha)
-\]
-sind festgelegt durch Wahl von $f(\alpha)$.
-
-\begin{itemize}
-\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt
-\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein
-\end{itemize}
-
-\end{block}}
-\uncover<8->{%
-\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$}
-$\mathbb{Q}\subset \Bbbk$,
-$n$ verschiedene Nullstellen $\mathbb{C}$:
-\[
-\uncover<9->{
-\operatorname{Gal}(\Bbbk/\mathbb{Q})
-\cong
-S_n}
-\uncover<10->{
-\quad
-\text{auflösbar?}}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{\uncover<5->{Galois-Gruppen}}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{1.2}
-
-\uncover<2->{
-\fill[color=blue!20] (-1.1,-0.3)  rectangle (0.3,{5*\s+0.3});
-\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen};
-}
-
-\node at (0,0) {$\mathbb{Q}$};
-\node at (0,{1*\s}) {$E_1$};
-\node at (0,{2*\s}) {$E_2$};
-\node at (0,{3*\s}) {$E_3$};
-\node at (0,{4*\s}) {$\vdots\mathstrut$};
-\node at (0,{5*\s}) {$\Bbbk$};
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s});
-
-\begin{scope}[xshift=0.5cm]
-\uncover<7->{
-\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3});
-\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe};
-}
-\uncover<5->{
-\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$};
-\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$};
-\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$};
-\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$};
-\node at (1,{4*\s}) {$\vdots\mathstrut$};
-\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$};
-\node at (1,{0.5*\s}) {$\cap\mathstrut$};
-\node at (1,{1.5*\s}) {$\cap\mathstrut$};
-\node at (1,{2.5*\s}) {$\cap\mathstrut$};
-\node at (1,{3.5*\s}) {$\cap\mathstrut$};
-\node at (1,{4.5*\s}) {$\cap\mathstrut$};
-}
-
-\uncover<6->{
-\begin{scope}[xshift=2.5cm]
-\node at (0,{0*\s}) {$G_n$};
-\node at (0,{1*\s}) {$G_{n-1}$};
-\node at (0,{2*\s}) {$G_{n-2}$};
-\node at (0,{3*\s}) {$G_{n-3}$};
-\node at (0,{5*\s}) {$G_0=\{e\}$};
-\node at (0,{0.5*\s}) {$\cap\mathstrut$};
-\node at (0,{1.5*\s}) {$\cap\mathstrut$};
-\node at (0,{2.5*\s}) {$\cap\mathstrut$};
-\node at (0,{3.5*\s}) {$\cap\mathstrut$};
-\node at (0,{4.5*\s}) {$\cap\mathstrut$};
-}
-
-\uncover<7->{
-\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$};
-\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch};
-
-\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$};
-\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch};
-
-\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$};
-\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch};
-}
-
-\end{scope}
-\end{scope}
-
-
-
-\end{tikzpicture}
-\end{center}
-\end{block}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% aufloesbarkeit.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Auflösbarkeit}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Radikalerweiterung}
+Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$
+einer Radikalerweiterung
+\[
+\Bbbk \subset \Bbbk(\alpha)
+\]
+sind festgelegt durch Wahl von $f(\alpha)$.
+
+\begin{itemize}
+\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt
+\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein
+\end{itemize}
+
+\end{block}}
+\uncover<8->{%
+\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$}
+$\mathbb{Q}\subset \Bbbk$,
+$n$ verschiedene Nullstellen $\mathbb{C}$:
+\[
+\uncover<9->{
+\operatorname{Gal}(\Bbbk/\mathbb{Q})
+\cong
+S_n}
+\uncover<10->{
+\quad
+\text{auflösbar?}}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{\uncover<5->{Galois-Gruppen}}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\s{1.2}
+
+\uncover<2->{
+\fill[color=blue!20] (-1.1,-0.3)  rectangle (0.3,{5*\s+0.3});
+\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen};
+}
+
+\node at (0,0) {$\mathbb{Q}$};
+\node at (0,{1*\s}) {$E_1$};
+\node at (0,{2*\s}) {$E_2$};
+\node at (0,{3*\s}) {$E_3$};
+\node at (0,{4*\s}) {$\vdots\mathstrut$};
+\node at (0,{5*\s}) {$\Bbbk$};
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s});
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s});
+
+\begin{scope}[xshift=0.5cm]
+\uncover<7->{
+\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3});
+\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe};
+}
+\uncover<5->{
+\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$};
+\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$};
+\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$};
+\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$};
+\node at (1,{4*\s}) {$\vdots\mathstrut$};
+\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$};
+\node at (1,{0.5*\s}) {$\cap\mathstrut$};
+\node at (1,{1.5*\s}) {$\cap\mathstrut$};
+\node at (1,{2.5*\s}) {$\cap\mathstrut$};
+\node at (1,{3.5*\s}) {$\cap\mathstrut$};
+\node at (1,{4.5*\s}) {$\cap\mathstrut$};
+}
+
+\uncover<6->{
+\begin{scope}[xshift=2.5cm]
+\node at (0,{0*\s}) {$G_n$};
+\node at (0,{1*\s}) {$G_{n-1}$};
+\node at (0,{2*\s}) {$G_{n-2}$};
+\node at (0,{3*\s}) {$G_{n-3}$};
+\node at (0,{5*\s}) {$G_0=\{e\}$};
+\node at (0,{0.5*\s}) {$\cap\mathstrut$};
+\node at (0,{1.5*\s}) {$\cap\mathstrut$};
+\node at (0,{2.5*\s}) {$\cap\mathstrut$};
+\node at (0,{3.5*\s}) {$\cap\mathstrut$};
+\node at (0,{4.5*\s}) {$\cap\mathstrut$};
+}
+
+\uncover<7->{
+\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$};
+\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch};
+
+\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$};
+\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch};
+
+\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$};
+\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch};
+}
+
+\end{scope}
+\end{scope}
+
+
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
-- 
cgit v1.2.1

