<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%
% euklidpoly.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Euklidischer Algorithmus in $\mathbb{F}_2[X]$}
Gegeben: $m(X)=X^4+X+1$, $b(X) = {\color{blue}X^2+1}$
\\
\uncover&lt;2-&gt;{Berechne $s,t\in\mathbb{F}_2[X]$ derart, dass $sm+tb=1$}
\uncover&lt;3-&gt;{%
\begin{center}
\begin{tabular}{|&gt;{$}c&lt;{$}|&gt;{$}c&lt;{$}&gt;{$}c&lt;{$}|&gt;{$}c&lt;{$}|&gt;{$}c&lt;{$}|&gt;{$}c&lt;{$}&gt;{$}c&lt;{$}|}
\hline
k&amp;    a_k&amp;  b_k&amp;  q_k&amp;r_k&amp;  c_k&amp;    d_k\\
\hline
 &amp;       &amp;     &amp;     &amp;   &amp;    1&amp;      0\\
0&amp;X^4+X+1&amp;{\color{blue}X^2+1}&amp;\uncover&lt;4-&gt;{X^2+1}&amp;\uncover&lt;4-&gt;{X}&amp;    0&amp;      1\\
1&amp;\uncover&lt;5-&gt;{X^2+1 }&amp;\uncover&lt;5-&gt;{X}&amp;\uncover&lt;5-&gt;{X}&amp;\uncover&lt;5-&gt;{1}&amp;\uncover&lt;5-&gt;{1}&amp;\uncover&lt;5-&gt;{X^2+1}\\
2&amp;\uncover&lt;6-&gt;{X     }&amp;\uncover&lt;6-&gt;{1}&amp;\uncover&lt;6-&gt;{X}&amp;\uncover&lt;6-&gt;{0}&amp;\uncover&lt;6-&gt;{{\color{red}X}}&amp;\uncover&lt;6-&gt;{{\color{red}X^3+X+1}}\\
3&amp;\uncover&lt;7-&gt;{1     }&amp;\uncover&lt;7-&gt;{0}&amp;&amp;&amp;\uncover&lt;7-&gt;{X^2+1}&amp;\uncover&lt;7-&gt;{X^4+X+1} \\
\hline
\end{tabular}
\end{center}}
\ifthenelse{\boolean{presentation}}{
\only&lt;8-&gt;{%
\begin{block}{Kontrolle}
\vspace{-10pt}
\begin{align*}
{\color{red}X}\cdot (X^4+X+1) + ({\color{red}X^3+X+1})({\color{blue}X^2+1})
&amp;\uncover&lt;9-&gt;{=
(X^5+X^2+X)}\\
&amp;\qquad \uncover&lt;10-&gt;{+ (X^5+X^3+X^2+X^3+X+1)}
\\
&amp;\uncover&lt;11-&gt;{=(X^5+X^2+X) + (X^5+X^2+X+1)}
\\
&amp;\uncover&lt;12-&gt;{=1}
\end{align*}
\end{block}}}{}
\begin{block}{Rechenregeln in $\mathbb{F}_2$}
$1+1=0$,
$2=0$, $+1=-1$.
\end{block}

\end{frame}
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