update to current state of book

13 files changed, 949 insertions, 949 deletions

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}

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}

diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex
index 6909849..20b278e 100644
--- a/vorlesungen/slides/4/galois/erweiterung.tex
+++ b/vorlesungen/slides/4/galois/erweiterung.tex
@@ -1,65 +1,65 @@
-%
-% erweiterung.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Körpererweiterungen}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Körpererweiterung}
-$E,F$ Körper: $E\subset F$
-\end{block}
-\uncover<6->{%
-\begin{block}{Vektorraum}
-$F$ ist ein Vektorraum über $E$
-\end{block}}
-\uncover<7->{%
-\begin{block}{Endliche Körpererweiterung}
-$\dim_E F < \infty$
-\end{block}}
-\uncover<8->{%
-\begin{block}{Adjunktion eines $\alpha$}
-$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und
-$\alpha$ enthält.
-\end{block}}
-\uncover<9->{%
-\begin{block}{Algebraische Erweiterung}
-$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von
-$m(X)\in\Bbbk[X]$
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Beispiele}
-\begin{enumerate}
-\item<3->
-$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$
-\item<4->
-$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$
-\item<5->
-$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$
-\end{enumerate}
-\end{block}}
-\uncover<7->{%
-\begin{block}{Grad}
-$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls
-\[
-\dim_E F  = n =: [F:E]
-\]
-\uncover<8->{%
-Gleichbedeutend: $\deg m(X) = n$}
-\uncover<10->{%
-\[
-E\subset F\subset G
-\Rightarrow
-[G:E] = [G:F]\cdot [F:E]
-\]
-(in unseren Fällen)}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% erweiterung.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\setlength{\abovedisplayskip}{5pt}

+\setlength{\belowdisplayskip}{5pt}

+\frametitle{Körpererweiterungen}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{block}{Körpererweiterung}

+$E,F$ Körper: $E\subset F$

+\end{block}

+\uncover<6->{%

+\begin{block}{Vektorraum}

+$F$ ist ein Vektorraum über $E$

+\end{block}}

+\uncover<7->{%

+\begin{block}{Endliche Körpererweiterung}

+$\dim_E F < \infty$

+\end{block}}

+\uncover<8->{%

+\begin{block}{Adjunktion eines $\alpha$}

+$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und

+$\alpha$ enthält.

+\end{block}}

+\uncover<9->{%

+\begin{block}{Algebraische Erweiterung}

+$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von

+$m(X)\in\Bbbk[X]$

+\end{block}}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<2->{%

+\begin{block}{Beispiele}

+\begin{enumerate}

+\item<3->

+$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$

+\item<4->

+$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$

+\item<5->

+$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$

+\end{enumerate}

+\end{block}}

+\uncover<7->{%

+\begin{block}{Grad}

+$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls

+\[

+\dim_E F  = n =: [F:E]

+\]

+\uncover<8->{%

+Gleichbedeutend: $\deg m(X) = n$}

+\uncover<10->{%

+\[

+E\subset F\subset G

+\Rightarrow

+[G:E] = [G:F]\cdot [F:E]

+\]

+(in unseren Fällen)}

+\end{block}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile
index 444944e..fd197ce 100644
--- a/vorlesungen/slides/4/galois/images/Makefile
+++ b/vorlesungen/slides/4/galois/images/Makefile
@@ -1,12 +1,12 @@
-#
-# Makefile
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-all:	wuerfel2.png wuerfel.png
-
-wuerfel.png:	wuerfel.pov common.inc
-	povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov
-
-wuerfel2.png:	wuerfel2.pov common.inc
-	povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov
+#

+# Makefile

+#

+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+#

+all:	wuerfel2.png wuerfel.png

+

+wuerfel.png:	wuerfel.pov common.inc

+	povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov

+

+wuerfel2.png:	wuerfel2.pov common.inc

+	povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov

diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc
index 6cfcabe..44ee4c8 100644
--- a/vorlesungen/slides/4/galois/images/common.inc
+++ b/vorlesungen/slides/4/galois/images/common.inc
@@ -1,89 +1,89 @@
-//
-// common.inc
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-#include "textures.inc"
-#include "stones.inc"
-
-global_settings {
-	assumed_gamma 1
-}
-
-#declare imagescale = 0.133;
-#declare O = <0, 0, 0>;
-#declare E = <1, 1, 1>;
-#declare a = pow(2, 1/3);
-#declare at = 0.02;
-
-camera {
-	location <3, 2, 12>
-	look_at E * (a / 2) * 0.93
-	right x * imagescale
-	up y * imagescale
-}
-
-light_source {
-	<11, 20, 16> color White
-	area_light <1,0,0> <0,0,1>, 10, 10
-	adaptive 1
-	jitter
-}
-
-sky_sphere {
-	pigment {
-		color rgb<1,1,1>
-	}
-}
-
-#macro wuerfelgitter(A, AT)
-	cylinder { O, <A, 0, 0>, AT }
-	cylinder { O, <0, A, 0>, AT }
-	cylinder { O, <0, 0, A>, AT }
-	cylinder { <A, 0, 0>, <A, A, 0>, AT }
-	cylinder { <A, 0, 0>, <A, 0, A>, AT }
-	cylinder { <0, A, 0>, <A, A, 0>, AT }
-	cylinder { <0, A, 0>, <0, A, A>, AT }
-	cylinder { <0, 0, A>, <A, 0, A>, AT }
-	cylinder { <0, 0, A>, <0, A, A>, AT }
-	cylinder { <A, A, 0>, <A, A, A>, AT }
-	cylinder { <A, 0, A>, <A, A, A>, AT }
-	cylinder { <0, A, A>, <A, A, A>, AT }
-	sphere { <0, 0, 0>, AT }
-	sphere { <A, 0, 0>, AT }
-	sphere { <0, A, 0>, AT }
-	sphere { <0, 0, A>, AT }
-	sphere { <A, A, 0>, AT }
-	sphere { <A, 0, A>, AT }
-	sphere { <0, A, A>, AT }
-	sphere { <A, A, A>, AT }
-#end
-
-#macro wuerfel()
-	union {
-		box { O, E }
-		wuerfelgitter(1, 0.5*at)
-		texture {
-			T_Grnt24
-		}
-		finish {
-			specular 0.9
-			metallic
-		}
-	}
-#end
-
-#macro wuerfel2()
-	union {
-		wuerfelgitter(a, at)
-		pigment {
-			color rgb<0.8,0.4,0.4>
-		}
-		finish {
-			specular 0.9
-			metallic
-		}
-	}
-#end
+//

+// common.inc

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#version 3.7;

+#include "colors.inc"

+#include "textures.inc"

+#include "stones.inc"

+

+global_settings {

+	assumed_gamma 1

+}

+

+#declare imagescale = 0.133;

+#declare O = <0, 0, 0>;

+#declare E = <1, 1, 1>;

+#declare a = pow(2, 1/3);

+#declare at = 0.02;

+

+camera {

+	location <3, 2, 12>

+	look_at E * (a / 2) * 0.93

+	right x * imagescale

+	up y * imagescale

+}

+

+light_source {

+	<11, 20, 16> color White

+	area_light <1,0,0> <0,0,1>, 10, 10

+	adaptive 1

+	jitter

+}

+

+sky_sphere {

+	pigment {

+		color rgb<1,1,1>

+	}

+}

+

+#macro wuerfelgitter(A, AT)

+	cylinder { O, <A, 0, 0>, AT }

+	cylinder { O, <0, A, 0>, AT }

+	cylinder { O, <0, 0, A>, AT }

+	cylinder { <A, 0, 0>, <A, A, 0>, AT }

+	cylinder { <A, 0, 0>, <A, 0, A>, AT }

+	cylinder { <0, A, 0>, <A, A, 0>, AT }

+	cylinder { <0, A, 0>, <0, A, A>, AT }

+	cylinder { <0, 0, A>, <A, 0, A>, AT }

+	cylinder { <0, 0, A>, <0, A, A>, AT }

+	cylinder { <A, A, 0>, <A, A, A>, AT }

+	cylinder { <A, 0, A>, <A, A, A>, AT }

+	cylinder { <0, A, A>, <A, A, A>, AT }

+	sphere { <0, 0, 0>, AT }

+	sphere { <A, 0, 0>, AT }

+	sphere { <0, A, 0>, AT }

+	sphere { <0, 0, A>, AT }

+	sphere { <A, A, 0>, AT }

+	sphere { <A, 0, A>, AT }

+	sphere { <0, A, A>, AT }

+	sphere { <A, A, A>, AT }

+#end

+

+#macro wuerfel()

+	union {

+		box { O, E }

+		wuerfelgitter(1, 0.5*at)

+		texture {

+			T_Grnt24

+		}

+		finish {

+			specular 0.9

+			metallic

+		}

+	}

+#end

+

+#macro wuerfel2()

+	union {

+		wuerfelgitter(a, at)

+		pigment {

+			color rgb<0.8,0.4,0.4>

+		}

+		finish {

+			specular 0.9

+			metallic

+		}

+	}

+#end

diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov
index a5db465..a0466f3 100644
--- a/vorlesungen/slides/4/galois/images/wuerfel.pov
+++ b/vorlesungen/slides/4/galois/images/wuerfel.pov
@@ -1,9 +1,9 @@
-//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-
+//

+// wuerfel.pov

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#include "common.inc"

+

+wuerfel()

+

diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov
index ac32b2f..a11bab0 100644
--- a/vorlesungen/slides/4/galois/images/wuerfel2.pov
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov
@@ -1,9 +1,9 @@
-//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-wuerfel2()
+//

+// wuerfel.pov

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#include "common.inc"

+

+wuerfel()

+wuerfel2()

diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex
index 094b570..b461d44 100644
--- a/vorlesungen/slides/4/galois/konstruktion.tex
+++ b/vorlesungen/slides/4/galois/konstruktion.tex
@@ -1,147 +1,147 @@
-%
-% konstruktion.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Konstruktion mit Zirkel und Lineal}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Strahlensatz}
-\uncover<6->{%
-Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$
-kann mit Zirkel und Lineal konstruiert werden.}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<7->{%
-\begin{block}{Kreis--Gerade}
-Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$
-\uncover<13->{%
-$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}
-\end{block}}
-\end{column}
-\end{columns}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{0.5}
-\def\t{0.45}
-
-\coordinate (A) at (0,0);
-\coordinate (B) at ({10*\t},0);
-
-\uncover<2->{
-	\draw (0,0) -- (30:{10.5*\s});
-}
-
-\uncover<3->{
-	\foreach \x in {0,...,10}{
-		\fill (30:{\x*\s}) circle[radius=0.03];
-	}
-	\foreach \x in {0,1,2,3,4,7,8,9}{
-		\node at (30:{\x*\s}) [above] {\tiny $\x$};
-	}
-	\node at (30:{10*\s}) [above right] {$q=10$};
-}
-
-\uncover<4->{
-	\foreach \x in {1,...,10}{
-		\fill (0:{\x*\t}) circle[radius=0.03];
-		\draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});
-	}
-}
-
-\draw (A) -- (0:{10.5*\t});
-\node at (A) [below left] {$A$};
-\node at (B) [below right] {$B$};
-\fill (A) circle[radius=0.05];
-\fill (B) circle[radius=0.05];
-
-\uncover<5->{
-	\node at (30:{6*\s}) [above left] {$p=6$};
-	\draw[line width=0.2pt] (0,0) -- (0,-0.4);
-	\draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);
-	\draw[<->] (0,-0.3) -- ({6*\t},-0.3);
-	\node at ({3*\t},-0.4) [below]
-		{$\displaystyle\frac{p}{q}\cdot\overline{AB}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<8->{%
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-
-%\foreach \x in {8,...,14}{
-%	\only<\x>{\node at (4,4) {$\x$};}
-%}
-
-\def\r{4}
-\def\a{50}
-
-\coordinate (A) at ({\r*cos(\a)},0);
-
-\uncover<10->{
-	\fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;
-	\fill[color=gray]
-		(95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;
-}
-
-\draw[->] (0,0) -- (95:\r);
-\node at (95:{0.5*\r}) [left] {$c$};
-
-\begin{scope}
-	\clip (-1,-0.3) rectangle (4.5,4.1);
-	\uncover<10->{
-		\draw (-1,0) -- (5,0);
-		\draw[->] (0,0) -- (\r,0);
-		\draw (0,0) circle[radius=\r];
-		\draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);
-	}
-\end{scope}
-
-\uncover<11->{
-	\fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;
-}
-
-\uncover<9->{
-	\fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;
-	\fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];
-	\draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);
-}
-
-\uncover<11->{
-	\draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);
-	\node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};
-}
-
-\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);
-\fill[color=blue] (0,0) circle[radius=0.04];
-\fill[color=blue] (A) circle[radius=0.04];
-\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};
-
-\uncover<12->{
-	\fill[color=white,opacity=0.8]
-		({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})
-		rectangle
-		({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});
-
-	\node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]
-		{$b=\sqrt{c^2-a^2}$};
-	\draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);
-	\fill[color=red] (\a:\r) circle[radius=0.05];
-	\fill[color=red] (A) circle[radius=0.05];
-}
-
-\end{tikzpicture}
-\end{center}}
-\end{column}
-\end{columns}
-\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:}
-Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$}
-\end{frame}
+%

+% konstruktion.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\frametitle{Konstruktion mit Zirkel und Lineal}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{block}{Strahlensatz}

+\uncover<6->{%

+Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$

+kann mit Zirkel und Lineal konstruiert werden.}

+\end{block}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<7->{%

+\begin{block}{Kreis--Gerade}

+Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$

+\uncover<13->{%

+$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}

+\end{block}}

+\end{column}

+\end{columns}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\def\s{0.5}

+\def\t{0.45}

+

+\coordinate (A) at (0,0);

+\coordinate (B) at ({10*\t},0);

+

+\uncover<2->{

+	\draw (0,0) -- (30:{10.5*\s});

+}

+

+\uncover<3->{

+	\foreach \x in {0,...,10}{

+		\fill (30:{\x*\s}) circle[radius=0.03];

+	}

+	\foreach \x in {0,1,2,3,4,7,8,9}{

+		\node at (30:{\x*\s}) [above] {\tiny $\x$};

+	}

+	\node at (30:{10*\s}) [above right] {$q=10$};

+}

+

+\uncover<4->{

+	\foreach \x in {1,...,10}{

+		\fill (0:{\x*\t}) circle[radius=0.03];

+		\draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});

+	}

+}

+

+\draw (A) -- (0:{10.5*\t});

+\node at (A) [below left] {$A$};

+\node at (B) [below right] {$B$};

+\fill (A) circle[radius=0.05];

+\fill (B) circle[radius=0.05];

+

+\uncover<5->{

+	\node at (30:{6*\s}) [above left] {$p=6$};

+	\draw[line width=0.2pt] (0,0) -- (0,-0.4);

+	\draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);

+	\draw[<->] (0,-0.3) -- ({6*\t},-0.3);

+	\node at ({3*\t},-0.4) [below]

+		{$\displaystyle\frac{p}{q}\cdot\overline{AB}$};

+}

+

+\end{tikzpicture}

+\end{center}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<8->{%

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+

+%\foreach \x in {8,...,14}{

+%	\only<\x>{\node at (4,4) {$\x$};}

+%}

+

+\def\r{4}

+\def\a{50}

+

+\coordinate (A) at ({\r*cos(\a)},0);

+

+\uncover<10->{

+	\fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;

+	\fill[color=gray]

+		(95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;

+}

+

+\draw[->] (0,0) -- (95:\r);

+\node at (95:{0.5*\r}) [left] {$c$};

+

+\begin{scope}

+	\clip (-1,-0.3) rectangle (4.5,4.1);

+	\uncover<10->{

+		\draw (-1,0) -- (5,0);

+		\draw[->] (0,0) -- (\r,0);

+		\draw (0,0) circle[radius=\r];

+		\draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);

+	}

+\end{scope}

+

+\uncover<11->{

+	\fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;

+}

+

+\uncover<9->{

+	\fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;

+	\fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];

+	\draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);

+}

+

+\uncover<11->{

+	\draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);

+	\node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};

+}

+

+\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);

+\fill[color=blue] (0,0) circle[radius=0.04];

+\fill[color=blue] (A) circle[radius=0.04];

+\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};

+

+\uncover<12->{

+	\fill[color=white,opacity=0.8]

+		({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})

+		rectangle

+		({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});

+

+	\node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]

+		{$b=\sqrt{c^2-a^2}$};

+	\draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);

+	\fill[color=red] (\a:\r) circle[radius=0.05];

+	\fill[color=red] (A) circle[radius=0.05];

+}

+

+\end{tikzpicture}

+\end{center}}

+\end{column}

+\end{columns}

+\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:}

+Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex
index f5763b9..f9510ba 100644
--- a/vorlesungen/slides/4/galois/quadratur.tex
+++ b/vorlesungen/slides/4/galois/quadratur.tex
@@ -1,66 +1,66 @@
-%
-% quadratur.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Quadratur des Kreises}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.44\textwidth}
-\begin{center}
-\uncover<2->{%
-\begin{tikzpicture}[>=latex,thick]
-
-\def\r{2.8}
-\pgfmathparse{sqrt(3.14159)*\r/2}
-\xdef\s{\pgfmathresult}
-
-\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);
-\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];
-
-\uncover<3->{
-	\draw[->,color=red] (0,0) -- (50:\r);
-	\fill[color=red] (0,0) circle[radius=0.04];
-	\node[color=red] at (50:{0.5*\r}) [below right] {$r$};
-}
-
-\uncover<4->{
-	\draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});
-	\draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});
-	\draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});
-	\node[color=blue] at (0,{-\s-0.6}) [below] {$l$};
-}
-
-\uncover<5->{
-	\node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}
-		\;\Rightarrow\;
-		{\color{blue}l}={\color{red}\sqrt{\pi}r}$};
-}
-
-\end{tikzpicture}}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere ein zu einem Kreis flächengleiches Quadrat
-\end{block}
-\uncover<6->{%
-\begin{block}{Modifizierte Aufgabe}
-Konstruiere eine Strecke, deren Länge Lösung der Gleichung
-$x^2-\pi=0$ ist.
-\end{block}}
-\uncover<7->{%
-\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]
-\begin{itemize}
-\item<8-> Lösung in einem Erweiterungskörper
-\item<9-> Lösung ist Nullstelle eines Polynoms 
-\item<10-> Lösung ist algebraisch
-\item<11-> $\pi$ ist {\bf nicht} algebraisch
-\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% quadratur.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\frametitle{Quadratur des Kreises}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.44\textwidth}

+\begin{center}

+\uncover<2->{%

+\begin{tikzpicture}[>=latex,thick]

+

+\def\r{2.8}

+\pgfmathparse{sqrt(3.14159)*\r/2}

+\xdef\s{\pgfmathresult}

+

+\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);

+\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];

+

+\uncover<3->{

+	\draw[->,color=red] (0,0) -- (50:\r);

+	\fill[color=red] (0,0) circle[radius=0.04];

+	\node[color=red] at (50:{0.5*\r}) [below right] {$r$};

+}

+

+\uncover<4->{

+	\draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});

+	\draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});

+	\draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});

+	\node[color=blue] at (0,{-\s-0.6}) [below] {$l$};

+}

+

+\uncover<5->{

+	\node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}

+		\;\Rightarrow\;

+		{\color{blue}l}={\color{red}\sqrt{\pi}r}$};

+}

+

+\end{tikzpicture}}

+\end{center}

+\end{column}

+\begin{column}{0.52\textwidth}

+\begin{block}{Aufgabe}

+Konstruiere ein zu einem Kreis flächengleiches Quadrat

+\end{block}

+\uncover<6->{%

+\begin{block}{Modifizierte Aufgabe}

+Konstruiere eine Strecke, deren Länge Lösung der Gleichung

+$x^2-\pi=0$ ist.

+\end{block}}

+\uncover<7->{%

+\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]

+\begin{itemize}

+\item<8-> Lösung in einem Erweiterungskörper

+\item<9-> Lösung ist Nullstelle eines Polynoms 

+\item<10-> Lösung ist algebraisch

+\item<11-> $\pi$ ist {\bf nicht} algebraisch

+\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}

+\qedhere

+\end{itemize}

+\end{proof}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex
index e9e4ce8..cb08dca 100644
--- a/vorlesungen/slides/4/galois/radikale.tex
+++ b/vorlesungen/slides/4/galois/radikale.tex
@@ -1,69 +1,69 @@
-%
-% radikale.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Lösung durch Radikale}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Problemstellung}
-Finde Nullstellen eines Polynomes
-\[
-p(X)
-=
-a_nX^n + a_{n-1}X^{n-1}
-+\dots+
-a_1X+a_0
-\]
-$p\in\mathbb{Q}[X]$
-\end{block}
-\uncover<2->{%
-\begin{block}{Radikale}
-Geschachtelte Wurzelausdrücke
-\[
-\sqrt[3]{
--\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
-}
-+
-\sqrt[3]{
--\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
-}
-\]
-\uncover<3->{(Lösung von $x^3+px+q=0$)}
-\end{block}}
-\uncover<4->{%
-\begin{block}{Lösbar durch Radikale}
-Nullstelle von $p(X)$ ist ein Radikal
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<5->{%
-\begin{block}{Algebraische Formulierung}
-Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$,
-finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart,
-dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{:
-$\Bbbk = \mathbb{Q}[X]/(p)$}
-\end{block}}
-\uncover<7->{%
-\begin{block}{Radikalerweiterung}
-Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften
-\begin{itemize}
-\item<8-> $\alpha$ ist eine Einheitswurzel
-\item<9-> $\alpha^k\in\Bbbk$
-\end{itemize}
-\end{block}}
-\vspace{-5pt}
-\uncover<10->{%
-\begin{block}{Lösbar durch Radikale}
-Radikalerweiterungen
-\[
-\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha
-\]
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% radikale.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\setlength{\abovedisplayskip}{5pt}

+\setlength{\belowdisplayskip}{5pt}

+\frametitle{Lösung durch Radikale}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{block}{Problemstellung}

+Finde Nullstellen eines Polynomes

+\[

+p(X)

+=

+a_nX^n + a_{n-1}X^{n-1}

++\dots+

+a_1X+a_0

+\]

+$p\in\mathbb{Q}[X]$

+\end{block}

+\uncover<2->{%

+\begin{block}{Radikale}

+Geschachtelte Wurzelausdrücke

+\[

+\sqrt[3]{

+-\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}

+}

++

+\sqrt[3]{

+-\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}

+}

+\]

+\uncover<3->{(Lösung von $x^3+px+q=0$)}

+\end{block}}

+\uncover<4->{%

+\begin{block}{Lösbar durch Radikale}

+Nullstelle von $p(X)$ ist ein Radikal

+\end{block}}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<5->{%

+\begin{block}{Algebraische Formulierung}

+Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$,

+finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart,

+dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{:

+$\Bbbk = \mathbb{Q}[X]/(p)$}

+\end{block}}

+\uncover<7->{%

+\begin{block}{Radikalerweiterung}

+Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften

+\begin{itemize}

+\item<8-> $\alpha$ ist eine Einheitswurzel

+\item<9-> $\alpha^k\in\Bbbk$

+\end{itemize}

+\end{block}}

+\vspace{-5pt}

+\uncover<10->{%

+\begin{block}{Lösbar durch Radikale}

+Radikalerweiterungen

+\[

+\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha

+\]

+\end{block}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex
index 1cae3fa..f340825 100644
--- a/vorlesungen/slides/4/galois/sn.tex
+++ b/vorlesungen/slides/4/galois/sn.tex
@@ -1,87 +1,87 @@
-%
-% sn.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Nichtauflösbarkeit von $S_n$}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Die symmetrische Gruppe $S_n$}
-Permutationen auf $n$ Elementen
-\[
-\sigma
-=
-\begin{pmatrix}
-1&2&3&\dots&n\\
-\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n)
-\end{pmatrix}
-\]
-\end{block}
-\vspace{-10pt}
-\uncover<2->{%
-\begin{block}{Signum}
-$t(\sigma)=\mathstrut$ Anzahl Transpositionen
-\[
-\operatorname{sgn}(\sigma)
-=
-(-1)^{t(\sigma)}
-=
-\begin{cases}
-\phantom{-}1&\text{$t(\sigma)$ gerade}
-\\
--1&\text{$t(\sigma)$ ungerade}
-\end{cases}
-\]
-Homomorphismus!
-\end{block}}
-\uncover<3->{%
-\begin{block}{Die alternierende Gruppe $A_n$}
-\vspace{-12pt}
-\[
-A_n = \ker \operatorname{sgn}
-=
-\{\sigma\in  S_n\;|\;\operatorname{sgn}(\sigma)=1\}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<4->{%
-\begin{block}{Normale Untergruppe}
-\begin{itemize}
-\item
-$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$
-\item
-$G/N$ ist wohldefiniert
-\end{itemize}
-\end{block}}
-\vspace{-10pt}
-\uncover<5->{%
-\begin{block}{Einfache Gruppe}
-$G$ einfach $\Leftrightarrow$
-\[
-H\triangleleft G
-\;
-\Rightarrow
-\;
-\text{$H=\{e\}$ oder $H=G$}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<6->{%
-\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$}
-\begin{enumerate}
-\item<7-> Zeigen, dass $A_5$ einfach ist
-\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach
-\end{enumerate}
-\uncover<9->{%
-$\Rightarrow$ i.~A.~keine Lösung der 
-einer Polynomgleichung vom Grad $\ge 5$ durch Radikale
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% sn.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\setlength{\abovedisplayskip}{5pt}

+\setlength{\belowdisplayskip}{5pt}

+\frametitle{Nichtauflösbarkeit von $S_n$}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{block}{Die symmetrische Gruppe $S_n$}

+Permutationen auf $n$ Elementen

+\[

+\sigma

+=

+\begin{pmatrix}

+1&2&3&\dots&n\\

+\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n)

+\end{pmatrix}

+\]

+\end{block}

+\vspace{-10pt}

+\uncover<2->{%

+\begin{block}{Signum}

+$t(\sigma)=\mathstrut$ Anzahl Transpositionen

+\[

+\operatorname{sgn}(\sigma)

+=

+(-1)^{t(\sigma)}

+=

+\begin{cases}

+\phantom{-}1&\text{$t(\sigma)$ gerade}

+\\

+-1&\text{$t(\sigma)$ ungerade}

+\end{cases}

+\]

+Homomorphismus!

+\end{block}}

+\uncover<3->{%

+\begin{block}{Die alternierende Gruppe $A_n$}

+\vspace{-12pt}

+\[

+A_n = \ker \operatorname{sgn}

+=

+\{\sigma\in  S_n\;|\;\operatorname{sgn}(\sigma)=1\}

+\]

+\end{block}}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<4->{%

+\begin{block}{Normale Untergruppe}

+\begin{itemize}

+\item

+$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$

+\item

+$G/N$ ist wohldefiniert

+\end{itemize}

+\end{block}}

+\vspace{-10pt}

+\uncover<5->{%

+\begin{block}{Einfache Gruppe}

+$G$ einfach $\Leftrightarrow$

+\[

+H\triangleleft G

+\;

+\Rightarrow

+\;

+\text{$H=\{e\}$ oder $H=G$}

+\]

+\end{block}}

+\vspace{-10pt}

+\uncover<6->{%

+\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$}

+\begin{enumerate}

+\item<7-> Zeigen, dass $A_5$ einfach ist

+\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach

+\end{enumerate}

+\uncover<9->{%

+$\Rightarrow$ i.~A.~keine Lösung der 

+einer Polynomgleichung vom Grad $\ge 5$ durch Radikale

+}

+\end{block}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
index 54b941b..28c07fe 100644
--- a/vorlesungen/slides/4/galois/winkeldreiteilung.tex
+++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
@@ -1,94 +1,94 @@
-%
-% winkeldreiteilung.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Winkeldreiteilung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.43\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\r{5}
-\def\a{25}
-
-\uncover<3->{
-	\draw[line width=0.7pt] (\r,0) arc (0:90:\r);
-}
-
-\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;
-\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};
-
-\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);
-
-\uncover<2->{
-	\fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;
-	\draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);
-	\node[color=red] at ({0.5*\a}:{0.7*\r})
-		{$\displaystyle\frac{\alpha}{3}$};
-}
-
-\uncover<3->{
-	\fill[color=blue] ({3*\a}:\r) circle[radius=0.05];
-	\draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);
-
-	\fill[color=red] ({\a}:\r) circle[radius=0.05];
-	\draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);
-
-	\draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];
-	\draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];
-}
-
-
-\uncover<4->{
-\node at ({0.5*\r},-0.5) [below] {$\displaystyle
-\cos{\color{blue}\alpha}
-=
-4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}
-$};
-}
-
-\uncover<5->{
-	\node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};
-	\node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.53\textwidth}
-\begin{block}{Aufgabe}
-Teile einen Winkel in drei gleiche Teile
-\end{block}
-\vspace{-2pt}
-\uncover<6->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere $x$ aus $a$ derart, dass
-\[
-p(x)
-=
-x^3-\frac34 x -a = 0
-\]
-\uncover<7->{%
-$a=0$:}
-\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}
-\end{block}}
-\vspace{-2pt}
-\uncover<10->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<11->
-$a\ne 0$ $\Rightarrow$  $p(x)$ irreduzibel
-\item<12->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<13->
-Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% winkeldreiteilung.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\setlength{\abovedisplayskip}{5pt}

+\setlength{\belowdisplayskip}{5pt}

+\frametitle{Winkeldreiteilung}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.43\textwidth}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\def\r{5}

+\def\a{25}

+

+\uncover<3->{

+	\draw[line width=0.7pt] (\r,0) arc (0:90:\r);

+}

+

+\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;

+\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};

+

+\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);

+

+\uncover<2->{

+	\fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;

+	\draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);

+	\node[color=red] at ({0.5*\a}:{0.7*\r})

+		{$\displaystyle\frac{\alpha}{3}$};

+}

+

+\uncover<3->{

+	\fill[color=blue] ({3*\a}:\r) circle[radius=0.05];

+	\draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);

+

+	\fill[color=red] ({\a}:\r) circle[radius=0.05];

+	\draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);

+

+	\draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];

+	\draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];

+}

+

+

+\uncover<4->{

+\node at ({0.5*\r},-0.5) [below] {$\displaystyle

+\cos{\color{blue}\alpha}

+=

+4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}

+$};

+}

+

+\uncover<5->{

+	\node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};

+	\node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};

+}

+

+\end{tikzpicture}

+\end{center}

+\end{column}

+\begin{column}{0.53\textwidth}

+\begin{block}{Aufgabe}

+Teile einen Winkel in drei gleiche Teile

+\end{block}

+\vspace{-2pt}

+\uncover<6->{%

+\begin{block}{Algebraisierte Aufgabe}

+Konstruiere $x$ aus $a$ derart, dass

+\[

+p(x)

+=

+x^3-\frac34 x -a = 0

+\]

+\uncover<7->{%

+$a=0$:}

+\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}

+\end{block}}

+\vspace{-2pt}

+\uncover<10->{%

+\begin{proof}[Unmöglichkeitsbeweis]

+\begin{itemize}

+\item<11->

+$a\ne 0$ $\Rightarrow$  $p(x)$ irreduzibel

+\item<12->

+$p(x)$ definiert eine Körpererweiterung vom Grad $3$

+\item<13->

+Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$

+\qedhere

+\end{itemize}

+\end{proof}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex
index ada6079..907d60a 100644
--- a/vorlesungen/slides/4/galois/wuerfel.tex
+++ b/vorlesungen/slides/4/galois/wuerfel.tex
@@ -1,64 +1,64 @@
-%
-% wuerfel.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Würfelverdoppelung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};
-\uncover<2->{
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};
-}
-
-\uncover<3->{
-	\draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);
-	\node[color=blue] at (0.75,-2.3) [above] {$a$};
-}
-
-\uncover<4->{
-	\begin{scope}[yshift=0.03cm]
-		\draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);
-		\draw[color=red] (2.85,-2.7) -- (2.85,-3.0);
-		\draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);
-	\end{scope}
-	\node[color=red] at (0.36,-2.9) [below] {$b$};
-}
-
-\uncover<5->{
-\node at (0,-4) {$
-	2{\color{blue}a}^3={\color{red}b}^3
-	\uncover<6->{\;\Rightarrow\;
-	\frac{b}{a} = \sqrt[3]{2}}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere einen Würfel mit doppeltem Volumen
-\end{block}
-\uncover<7->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere eine Nullstelle von $p(x)=x^3-2$
-\end{block}}
-\uncover<8->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<9->
-$p(x)$ irreduzibel
-\item<10->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<11->
-Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% wuerfel.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\frametitle{Würfelverdoppelung}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};

+\uncover<2->{

+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};

+}

+

+\uncover<3->{

+	\draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);

+	\node[color=blue] at (0.75,-2.3) [above] {$a$};

+}

+

+\uncover<4->{

+	\begin{scope}[yshift=0.03cm]

+		\draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);

+		\draw[color=red] (2.85,-2.7) -- (2.85,-3.0);

+		\draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);

+	\end{scope}

+	\node[color=red] at (0.36,-2.9) [below] {$b$};

+}

+

+\uncover<5->{

+\node at (0,-4) {$

+	2{\color{blue}a}^3={\color{red}b}^3

+	\uncover<6->{\;\Rightarrow\;

+	\frac{b}{a} = \sqrt[3]{2}}$};

+}

+

+\end{tikzpicture}

+\end{center}

+\end{column}

+\begin{column}{0.52\textwidth}

+\begin{block}{Aufgabe}

+Konstruiere einen Würfel mit doppeltem Volumen

+\end{block}

+\uncover<7->{%

+\begin{block}{Algebraisierte Aufgabe}

+Konstruiere eine Nullstelle von $p(x)=x^3-2$

+\end{block}}

+\uncover<8->{%

+\begin{proof}[Unmöglichkeitsbeweis]

+\begin{itemize}

+\item<9->

+$p(x)$ irreduzibel

+\item<10->

+$p(x)$ definiert eine Körpererweiterung vom Grad $3$

+\item<11->

+Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar

+\qedhere

+\end{itemize}

+\end{proof}}

+\end{column}

+\end{columns}

+\end{frame}