Gauss

7 files changed, 155 insertions, 9 deletions

diff --git a/buch/chapters/30-endlichekoerper/uebungsaufgaben/3004/matrix.m b/buch/chapters/30-endlichekoerper/uebungsaufgaben/3004/matrix.m
index ed85185..42e9d9f 100644
--- a/buch/chapters/30-endlichekoerper/uebungsaufgaben/3004/matrix.m
+++ b/buch/chapters/30-endlichekoerper/uebungsaufgaben/3004/matrix.m
@@ -6,13 +6,14 @@
 
 n = 4
 N = 20;
+p = 2;
 
 d = 0;
 while d == 0
 	A = round(N * rand(n,n));
-	B = mod(A, 2);
+	B = mod(A, p);
 	d = det(B);
-	d = mod(d, 2);
+	d = mod(d, p);
 	d = d * B(1,1);
 end
 A
diff --git a/buch/chapters/30-endlichekoerper/wurzeln.tex b/buch/chapters/30-endlichekoerper/wurzeln.tex
index 5dc3fc2..02429dc 100644
--- a/buch/chapters/30-endlichekoerper/wurzeln.tex
+++ b/buch/chapters/30-endlichekoerper/wurzeln.tex
@@ -896,7 +896,7 @@ In Kapitel~\ref{buch:chapter:kryptographie} genauer untersucht.
 
 \subsection{Zerfällungskörper
 \label{buch:subsection:zerfaellungskoerper}}
-
+XXX TODO
 
 
 
diff --git a/vorlesungen/03_endlichekoerper/slides.tex b/vorlesungen/03_endlichekoerper/slides.tex
index 67c97d6..19a9ab0 100644
--- a/vorlesungen/03_endlichekoerper/slides.tex
+++ b/vorlesungen/03_endlichekoerper/slides.tex
@@ -4,12 +4,12 @@
 % (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
 \section{Euklidischer Algorithmus}
-% XXX \folie{4/ggt.tex}
-% XXX \folie{4/euklidmatrix.tex}
+\folie{4/ggt.tex}
+\folie{4/euklidmatrix.tex}
 
 \section{Die Galois-Körper $\mathbb{F}_p$}
-% XXX \folie{4/fp.tex}
-% XXX \folie{4/division.tex}
+\folie{4/fp.tex}
+\folie{4/division.tex}
 % XXX \folie{4/gauss.tex}
 % XXX \folie{4/dh.tex}
 % XXX ? \folie{4/polynomefp.tex}
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index 0adf913..91767a9 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -11,5 +11,6 @@ chapter4 =								\
 	../slides/4/euklidtabelle.tex					\
 	../slides/4/fp.tex						\
 	../slides/4/division.tex					\
+	../slides/4/gauss.tex						\
 	../slides/4/chapter.tex
 
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
index 1c23783..27884a1 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -9,3 +9,4 @@
 \folie{4/euklidtabelle.tex}
 \folie{4/fp.tex}
 \folie{4/division.tex}
+\folie{4/gauss.tex}
diff --git a/vorlesungen/slides/4/gauss.tex b/vorlesungen/slides/4/gauss.tex
new file mode 100644
index 0000000..960e8e1
--- /dev/null
+++ b/vorlesungen/slides/4/gauss.tex
@@ -0,0 +1,143 @@
+%
+% gauss.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\def\ds{0.5}
+\def\punkt#1#2{({(#1)*\ds},{-(#2)*\ds})}
+\def\tabelle{
+	\foreach \x in {-0.5,0.5,3.5}{
+		\draw \punkt{\x}{-0.5} -- \punkt{\x}{3.5};
+		\draw \punkt{-0.5}{\x} -- \punkt{3.5}{\x};
+	}
+	\node at \punkt{0}{1} {$0$};
+	\node at \punkt{0}{2} {$1$};
+	\node at \punkt{0}{3} {$2$};
+	\node at \punkt{1}{0} {$0$};
+	\node at \punkt{2}{0} {$1$};
+	\node at \punkt{3}{0} {$2$};
+}
+\begin{frame}[t]
+\frametitle{Gauss-Algorithmus in $\mathbb{F}_3$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Additions-/Multiplikationstabelle}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}[xshift=-1.6cm]
+\tabelle
+\node at \punkt{0}{0} {$+$};
+\node at \punkt{1}{1} {$0$};
+\node at \punkt{1}{2} {$1$};
+\node at \punkt{1}{3} {$2$};
+\node at \punkt{2}{1} {$1$};
+\node at \punkt{2}{2} {$2$};
+\node at \punkt{2}{3} {$0$};
+\node at \punkt{3}{1} {$2$};
+\node at \punkt{3}{2} {$0$};
+\node at \punkt{3}{3} {$1$};
+\end{scope}
+\begin{scope}[xshift=1.6cm]
+\tabelle
+\node at \punkt{0}{0} {$\cdot$};
+\node at \punkt{1}{1} {$0$};
+\node at \punkt{1}{2} {$0$};
+\node at \punkt{1}{3} {$0$};
+\node at \punkt{2}{1} {$0$};
+\node at \punkt{2}{2} {$1$};
+\node at \punkt{2}{3} {$2$};
+\node at \punkt{3}{1} {$0$};
+\node at \punkt{3}{2} {$2$};
+\node at \punkt{3}{3} {$1$};
+\end{scope}
+\end{tikzpicture}
+\end{center}
+
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<2->{%
+\begin{block}{Gleichungssystem\uncover<9->{/Lösung}}
+\[
+\left.
+\begin{array}{rcrcrcrcr}
+ x&+&y&+2z&=&1\\
+2x& & &+ z&=&2\\
+ x&+&y&   &=&2
+\end{array}
+\uncover<9->{
+\right\}
+\Rightarrow
+\left\{
+\begin{aligned}
+x&=2\\
+y&=0\\
+z&=1
+\end{aligned}
+\right.}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Gauss-Algorithmus}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\node at (0,0) {\begin{minipage}{13cm}%
+\[
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 2 & 1 \\
+ 2 & 0 & 1 & 2 \\
+ 1 & 1 & 0 & 2 \\
+\hline
+\end{tabular}
+\uncover<4->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 2 & 1 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\uncover<6->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 0 & 2 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\uncover<8->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 0 & 0 & 2 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\]
+\end{minipage}};
+\begin{scope}[yshift=0.2cm]
+\uncover<3->{
+\draw[color=red] (-5.6,0.3) circle[radius=0.2];
+\draw[color=blue] (-5.4,-0.8) -- (-5.4,-0.2) arc (0:180:0.2) -- (-5.8,-0.8);
+}
+\uncover<5->{
+\draw[color=blue] (-1.45,0.5) -- (-1.45,-0.2) arc (180:360:0.2) -- (-1.05,0.5);
+}
+\uncover<7->{
+\draw[color=blue] (1.05,0.5) -- (1.05,0.2) arc (180:360:0.2) -- (1.45,0.5);
+}
+\end{scope}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 05946f0..07fb6a8 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -30,7 +30,7 @@
 %\folie{4/euklidtabelle.tex}
 %\folie{4/fp.tex}
 %\folie{4/division.tex}
-% XXX \folie{4/gauss.tex}
+%\folie{4/gauss.tex}
 % XXX \folie{4/dh.tex}
 % XXX ? \folie{4/polynomefp.tex}
 % XXX \folie{4/frobenius.tex}
@@ -69,7 +69,7 @@
 %\folie{5/jordan.tex}
 % XXX Diagonalform
 % XXX \folie{5/diagonalform.tex}
-\folie{5/reellenormalform.tex}
+%\folie{5/reellenormalform.tex}
 % XXX \folie{5/hessenberg.tex}
 
 \section{Satz von Cayley-Hamilton}