diff options
Diffstat (limited to 'vorlesungen/slides/4/galois')
| -rw-r--r-- | vorlesungen/slides/4/galois/aufloesbarkeit.tex | 240 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/automorphismus.tex | 236 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/erweiterung.tex | 130 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/images/Makefile | 24 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/images/common.inc | 178 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/images/wuerfel.pov | 18 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/images/wuerfel2.pov | 18 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/konstruktion.tex | 294 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/quadratur.tex | 132 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/radikale.tex | 138 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/sn.tex | 174 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/winkeldreiteilung.tex | 188 | ||||
| -rw-r--r-- | vorlesungen/slides/4/galois/wuerfel.tex | 128 |
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}
|