diff options
Diffstat (limited to '')
| -rw-r--r-- | buch/chapters/30-endlichekoerper/uebungsaufgaben/3004/matrix.m | 5 | ||||
| -rw-r--r-- | buch/chapters/30-endlichekoerper/wurzeln.tex | 2 | ||||
| -rw-r--r-- | vorlesungen/03_endlichekoerper/slides.tex | 8 | ||||
| -rw-r--r-- | vorlesungen/slides/4/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/4/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/4/gauss.tex | 143 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 4 |
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} |