Merge pull request #1 from AndreasFMueller/master

update

1 files changed, 178 insertions, 0 deletions

diff --git a/vorlesungen/slides/4/fp.tex b/vorlesungen/slides/4/fp.tex
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+%
+% fp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\feld#1#2#3{
+	\node at ({#1},{5-#2}) {$#3$};
+}
+\def\geld#1#2#3{
+	\node at ({#1},{6-#2}) {$#3$};
+}
+\def\rot#1#2{
+	\fill[color=red!20] ({#1-0.5},{5-#2-0.5}) rectangle ({#1+0.5},{5-#2+0.5});
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\gruen#1#2{
+	\fill[color=darkgreen!20] ({#1-0.5},{6-#2-0.5}) rectangle ({#1+0.5},{6-#2+0.5});
+}
+\def\inverse#1#2{
+	\node at (9,{6-#1}) {$#1^{-1}=#2\mathstrut$};
+}
+\begin{frame}[t]
+\frametitle{Galois-Körper}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Restklassenring$\mathstrut$}
+$\mathbb{Z}/n\mathbb{Z}
+=\{ \llbracket r\rrbracket\;|\; 0\le r < n \} \mathstrut$
+ist ein Ring
+\end{block}
+\uncover<2->{%
+\begin{block}{Nullteiler}
+Falls $n=n_1n_2$, dann sind $\llbracket n_1\rrbracket$ und
+$\llbracket n_2\rrbracket$ Nullteiler in $\mathbb{Z}/n\mathbb{Z}$:
+\[
+\llbracket n_1\rrbracket
+\llbracket n_2\rrbracket
+=
+\llbracket n_1n_2 \rrbracket
+=
+\llbracket n\rrbracket
+=
+\llbracket 0 \rrbracket
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Galois-Körper $\mathbb{F}_p\mathstrut$}
+$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\mathstrut$
+\end{block}}
+\uncover<4->{%
+\begin{block}{$n$ prim}
+Für $n=p$ prim ist $\mathbb{Z}/n\mathbb{Z}$ nullteilerfrei
+\medskip
+
+\uncover<5->{
+$\Rightarrow \quad \mathbb{F}_p$ ist ein Körper
+}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.45]
+\fill[color=white] (-12,0) circle[radius=0.1];
+\fill[color=white] (12,0) circle[radius=0.1];
+\uncover<3->{
+\begin{scope}[xshift=-8cm]
+\rot{2}{3}
+\rot{4}{3}
+\rot{3}{2}
+\rot{3}{4}
+\fill[color=gray!40] (-0.5,5.5) rectangle (5.5,6.5);
+\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,5.5);
+\foreach \x in {-0.5,5.5}{
+	\draw (\x,-0.5) -- (\x,6.5);
+}
+\foreach \x in {0.5,...,4.5}{
+	\draw[line width=0.3pt] (\x,-0.5) -- (\x,6.5);
+}
+\foreach \y in {0.5,...,5.5}{
+	\draw[line width=0.3pt] (-1.5,\y) -- (5.5,\y);
+}
+\foreach \y in {-0.5,5.5}{
+	\draw (-1.5,\y) -- (5.5,\y);
+}
+\draw (-1.5,-0.5) -- (-1.5,5.5);
+\draw (-0.5,6.5) -- (5.5,6.5);
+\foreach \x in {0,...,5}{
+	\node at (\x,6) {$\x$};
+	\node at (-1,{5-\x}) {$\x$};
+}
+\foreach \x in {0,...,5}{
+	\feld{\x}{0}{0}
+	\feld{0}{\x}{0}
+}
+\foreach \x in {2,...,5}{
+	\feld{\x}{1}{\x}
+	\feld{1}{\x}{\x}
+}
+\feld{1}{1}{1}
+\feld{2}{2}{4}
+\feld{2}{3}{0} \feld{3}{2}{0}
+\feld{2}{4}{2} \feld{4}{2}{2}
+\feld{2}{5}{4} \feld{5}{2}{4}
+\feld{3}{3}{3}
+\feld{4}{3}{0} \feld{3}{4}{0}
+\feld{5}{3}{3} \feld{3}{5}{3}
+\feld{4}{4}{4}
+\feld{4}{5}{2} \feld{5}{4}{2}
+\feld{5}{5}{1}
+\end{scope}}
+\uncover<6->{
+\begin{scope}[xshift=6cm]
+\uncover<7->{ \gruen{1}{1} }
+\uncover<8->{ \gruen{4}{2} }
+\uncover<9->{ \gruen{5}{3} }
+\uncover<10->{ \gruen{2}{4} }
+\uncover<11->{ \gruen{3}{5} }
+\uncover<12->{ \gruen{6}{6} }
+\fill[color=gray!40] (-0.5,6.5) rectangle (6.5,7.5);
+\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,6.5);
+\foreach \x in {-0.5,6.5}{
+	\draw (\x,-0.5) -- (\x,7.5);
+}
+\foreach \x in {0.5,...,5.5}{
+	\draw[line width=0.3pt] (\x,-0.5) -- (\x,7.5);
+}
+\foreach \y in {0.5,...,6.5}{
+	\draw[line width=0.3pt] (-1.5,\y) -- (6.5,\y);
+}
+\foreach \y in {-0.5,6.5}{
+	\draw (-1.5,\y) -- (6.5,\y);
+}
+\draw (-1.5,-0.5) -- (-1.5,6.5);
+\draw (-0.5,7.5) -- (6.5,7.5);
+\foreach \x in {0,...,6}{
+	\node at (\x,7) {$\x$};
+	\node at (-1,{6-\x}) {$\x$};
+}
+\foreach \x in {0,...,6}{
+	\geld{\x}{0}{0}
+	\geld{0}{\x}{0}
+}
+\foreach \x in {2,...,6}{
+	\geld{\x}{1}{\x}
+	\geld{1}{\x}{\x}
+}
+\geld{1}{1}{1}
+\geld{2}{2}{4}
+\geld{2}{3}{6} \geld{3}{2}{6}
+\geld{2}{4}{1} \geld{4}{2}{1}
+\geld{2}{5}{3} \geld{5}{2}{3}
+\geld{2}{6}{5} \geld{6}{2}{5}
+\geld{3}{3}{2}
+\geld{4}{3}{5} \geld{3}{4}{5}
+\geld{5}{3}{1} \geld{3}{5}{1}
+\geld{6}{3}{4} \geld{3}{6}{4}
+\geld{4}{4}{2}
+\geld{5}{4}{6} \geld{4}{5}{6}
+\geld{6}{4}{3} \geld{4}{6}{3}
+\geld{5}{5}{4}
+\geld{6}{5}{2} \geld{5}{6}{2}
+\geld{6}{6}{1}
+\uncover<7->{ \inverse{1}{1} }
+\uncover<8->{ \inverse{2}{4} }
+\uncover<9->{ \inverse{3}{5} }
+\uncover<10->{ \inverse{4}{2} }
+\uncover<11->{ \inverse{5}{3} }
+\uncover<12->{ \inverse{6}{6} }
+\end{scope}}
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup