15 files changed, 949 insertions, 0 deletions
diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
new file mode 100644
index 0000000..3d52b00
--- /dev/null
+++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex
new file mode 100644
index 0000000..e59f9b9
--- /dev/null
+++ b/vorlesungen/slides/4/galois/automorphismus.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex
new file mode 100644
index 0000000..20b278e
--- /dev/null
+++ b/vorlesungen/slides/4/galois/erweiterung.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile
new file mode 100644
index 0000000..fd197ce
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/Makefile
@@ -0,0 +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
diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc
new file mode 100644
index 0000000..44ee4c8
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/common.inc
@@ -0,0 +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
diff --git a/vorlesungen/slides/4/galois/images/wuerfel.png b/vorlesungen/slides/4/galois/images/wuerfel.png
new file mode 100644
index 0000000..ff6fc14
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel.png
diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov
new file mode 100644
index 0000000..a0466f3
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel.pov
@@ -0,0 +1,9 @@
+//
+// wuerfel.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+wuerfel()
+
diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.png b/vorlesungen/slides/4/galois/images/wuerfel2.png
new file mode 100644
index 0000000..68919cc
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.png
diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov
new file mode 100644
index 0000000..a11bab0
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov
@@ -0,0 +1,9 @@
+//
+// 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
new file mode 100644
index 0000000..b461d44
--- /dev/null
+++ b/vorlesungen/slides/4/galois/konstruktion.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex
new file mode 100644
index 0000000..f9510ba
--- /dev/null
+++ b/vorlesungen/slides/4/galois/quadratur.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex
new file mode 100644
index 0000000..cb08dca
--- /dev/null
+++ b/vorlesungen/slides/4/galois/radikale.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex
new file mode 100644
index 0000000..f340825
--- /dev/null
+++ b/vorlesungen/slides/4/galois/sn.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
new file mode 100644
index 0000000..28c07fe
--- /dev/null
+++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
@@ -0,0 +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}
diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex
new file mode 100644
index 0000000..907d60a
--- /dev/null
+++ b/vorlesungen/slides/4/galois/wuerfel.tex
@@ -0,0 +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}