From 2db90bfe4b174570424c408f04000902411d8755 Mon Sep 17 00:00:00 2001 From: Joshua Baer Date: Mon, 12 Apr 2021 21:51:55 +0200 Subject: update to current state of book --- vorlesungen/slides/4/galois/aufloesbarkeit.tex | 240 +++++++++--------- vorlesungen/slides/4/galois/automorphismus.tex | 236 ++++++++--------- vorlesungen/slides/4/galois/erweiterung.tex | 130 +++++----- vorlesungen/slides/4/galois/images/Makefile | 24 +- vorlesungen/slides/4/galois/images/common.inc | 178 ++++++------- vorlesungen/slides/4/galois/images/wuerfel.pov | 18 +- vorlesungen/slides/4/galois/images/wuerfel2.pov | 18 +- vorlesungen/slides/4/galois/konstruktion.tex | 294 +++++++++++----------- vorlesungen/slides/4/galois/quadratur.tex | 132 +++++----- vorlesungen/slides/4/galois/radikale.tex | 138 +++++----- vorlesungen/slides/4/galois/sn.tex | 174 ++++++------- vorlesungen/slides/4/galois/winkeldreiteilung.tex | 188 +++++++------- vorlesungen/slides/4/galois/wuerfel.tex | 128 +++++----- 13 files changed, 949 insertions(+), 949 deletions(-) (limited to 'vorlesungen/slides/4/galois') 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, , AT } - cylinder { O, <0, A, 0>, AT } - cylinder { O, <0, 0, A>, AT } - cylinder { , , AT } - cylinder { , , AT } - cylinder { <0, A, 0>, , AT } - cylinder { <0, A, 0>, <0, A, A>, AT } - cylinder { <0, 0, A>, , AT } - cylinder { <0, 0, A>, <0, A, A>, AT } - cylinder { , , AT } - cylinder { , , AT } - cylinder { <0, A, A>, , AT } - sphere { <0, 0, 0>, AT } - sphere { , AT } - sphere { <0, A, 0>, AT } - sphere { <0, 0, A>, AT } - sphere { , AT } - sphere { , AT } - sphere { <0, A, A>, AT } - sphere { , 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, , AT } + cylinder { O, <0, A, 0>, AT } + cylinder { O, <0, 0, A>, AT } + cylinder { , , AT } + cylinder { , , AT } + cylinder { <0, A, 0>, , AT } + cylinder { <0, A, 0>, <0, A, A>, AT } + cylinder { <0, 0, A>, , AT } + cylinder { <0, 0, A>, <0, A, A>, AT } + cylinder { , , AT } + cylinder { , , AT } + cylinder { <0, A, A>, , AT } + sphere { <0, 0, 0>, AT } + sphere { , AT } + sphere { <0, A, 0>, AT } + sphere { <0, 0, A>, AT } + sphere { , AT } + sphere { , AT } + sphere { <0, A, A>, AT } + sphere { , 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} -- cgit v1.2.1