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/automorphismus.tex | 236 ++++++++++++------------- 1 file changed, 118 insertions(+), 118 deletions(-) (limited to 'vorlesungen/slides/4/galois/automorphismus.tex') 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} -- cgit v1.2.1