1 files changed, 118 insertions, 118 deletions
diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex
index 6051813..e59f9b9 100644
--- a/vorlesungen/slides/4/galois/automorphismus.tex
+++ b/vorlesungen/slides/4/galois/automorphismus.tex
@@ -1,118 +1,118 @@
-%
-% automorphismus.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{4pt}
-\setlength{\belowdisplayskip}{4pt}
-\frametitle{Galois-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.40\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{3.0}
-\begin{scope}[xshift=-1.5cm]
-\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut};
-\node at (0,{\s}) {$G$};
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0);
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0);
-\node at (0,{-0.5*\s}) [left] {$[F:E]$};
-\node at (0,{0.5*\s}) [left] {$[G:F]$};
-\node at (0,0) {$F$};
-\node at (0,{-\s}) {$E$};
-\end{scope}
-\uncover<3->{
-\begin{scope}[xshift=1.8cm]
-\node at (0,{\s+0.1}) [above] {Gruppe\strut};
-\fill (0,{-\s}) circle[radius=0.06];
-\fill (0,0) circle[radius=0.06];
-\fill (0,{\s}) circle[radius=0.06];
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
-	(0,{-\s}) to[out=100,in=-100] (0,{\s});
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
-	(0,{-\s}) to[out=80,in=-80] (0,0);
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
-	(0,0) to[out=80,in=-80] (0,{\s});
-\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$};
-\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$};
-\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$};
-\end{scope}
-\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s});
-\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$};
-}
-\uncover<4->{
-\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s});
-\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<2->{%
-\begin{block}{Automorphismus}
-\vspace{-10pt}
-\[
-\operatorname{Aut}(F)
-=
-\left\{
-f\colon F\to F
-\left|
-\begin{aligned}
-f(x+y)&=f(x)+f(y)\\
-f(xy)&=f(x)f(y)
-\end{aligned}
-\right.
-\right\}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<3->{%
-\begin{block}{Galois-Gruppe}
-Automorphismen, die $E$ festlassen
-\[
-{\color{red}
-\operatorname{Gal}(F/E)
-}
-=
-\left\{
-\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E
-\right\}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<4->{%
-\begin{block}{Fixkörper}
-$H\subset \operatorname{Aut}(F)$:
-\begin{align*}
-{\color{blue}F^H}
-&=
-\{x\in F\;|\; hx = x\forall h\in H\}
-=\operatorname{Fix}(H)
-\end{align*}
-\end{block}}
-\vspace{-13pt}
-\uncover<5->{%
-\begin{block}{Beispiel}
-\begin{itemize}
-\item<6->
-\(
-\operatorname{Gal}(\mathbb{C}/\mathbb{R})
-=
-\{
-\operatorname{id}_{\mathbb{C}},
-\operatorname{conj}\colon z\mapsto\overline{z}
-\}
-\)
-\item<7->
-\(
-\mathbb{C}^{\operatorname{conj}}
-=
-\mathbb{R}
-\)
-\end{itemize}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%
+% automorphismus.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{4pt}
+\setlength{\belowdisplayskip}{4pt}
+\frametitle{Galois-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\s{3.0}
+\begin{scope}[xshift=-1.5cm]
+\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut};
+\node at (0,{\s}) {$G$};
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0);
+\node at (0,{-0.5*\s}) [left] {$[F:E]$};
+\node at (0,{0.5*\s}) [left] {$[G:F]$};
+\node at (0,0) {$F$};
+\node at (0,{-\s}) {$E$};
+\end{scope}
+\uncover<3->{
+\begin{scope}[xshift=1.8cm]
+\node at (0,{\s+0.1}) [above] {Gruppe\strut};
+\fill (0,{-\s}) circle[radius=0.06];
+\fill (0,0) circle[radius=0.06];
+\fill (0,{\s}) circle[radius=0.06];
+\draw[shorten >= 0.1cm,shorten <= 0.1cm]
+	(0,{-\s}) to[out=100,in=-100] (0,{\s});
+\draw[shorten >= 0.1cm,shorten <= 0.1cm]
+	(0,{-\s}) to[out=80,in=-80] (0,0);
+\draw[shorten >= 0.1cm,shorten <= 0.1cm]
+	(0,0) to[out=80,in=-80] (0,{\s});
+\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$};
+\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$};
+\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$};
+\end{scope}
+\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s});
+\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$};
+}
+\uncover<4->{
+\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s});
+\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<2->{%
+\begin{block}{Automorphismus}
+\vspace{-10pt}
+\[
+\operatorname{Aut}(F)
+=
+\left\{
+f\colon F\to F
+\left|
+\begin{aligned}
+f(x+y)&=f(x)+f(y)\\
+f(xy)&=f(x)f(y)
+\end{aligned}
+\right.
+\right\}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{block}{Galois-Gruppe}
+Automorphismen, die $E$ festlassen
+\[
+{\color{red}
+\operatorname{Gal}(F/E)
+}
+=
+\left\{
+\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E
+\right\}
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<4->{%
+\begin{block}{Fixkörper}
+$H\subset \operatorname{Aut}(F)$:
+\begin{align*}
+{\color{blue}F^H}
+&=
+\{x\in F\;|\; hx = x\forall h\in H\}
+=\operatorname{Fix}(H)
+\end{align*}
+\end{block}}
+\vspace{-13pt}
+\uncover<5->{%
+\begin{block}{Beispiel}
+\begin{itemize}
+\item<6->
+\(
+\operatorname{Gal}(\mathbb{C}/\mathbb{R})
+=
+\{
+\operatorname{id}_{\mathbb{C}},
+\operatorname{conj}\colon z\mapsto\overline{z}
+\}
+\)
+\item<7->
+\(
+\mathbb{C}^{\operatorname{conj}}
+=
+\mathbb{R}
+\)
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}