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author | Joshua Baer <the.baer.joshua@gmail.ch> | 2021-04-12 21:51:55 +0200 |
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committer | Joshua Baer <the.baer.joshua@gmail.ch> | 2021-04-12 21:51:55 +0200 |
commit | 2db90bfe4b174570424c408f04000902411d8755 (patch) | |
tree | e297a6274ff748de27257bffd7097c6b362ba12d /vorlesungen/slides/4/galois/automorphismus.tex | |
parent | add new files (diff) | |
download | SeminarMatrizen-2db90bfe4b174570424c408f04000902411d8755.tar.gz SeminarMatrizen-2db90bfe4b174570424c408f04000902411d8755.zip |
update to current state of book
Diffstat (limited to 'vorlesungen/slides/4/galois/automorphismus.tex')
-rw-r--r-- | vorlesungen/slides/4/galois/automorphismus.tex | 236 |
1 files changed, 118 insertions, 118 deletions
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}
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