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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
\def \name {Valentin Brandl}					%
\def \matrikel {108018274494}				%
\def \pname {Marvin Herrmann}				%
\def \pmatrikel {108018265436}				%
\def \gruppe {2 (Mi 10-12 Andre)}
\def \qname {Pascal Brackmann}
\def \qmatrikel {108017113834}				%
\def \uebung {11}								%
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\newcommand{\hwsol}{
\vspace*{-2cm}
\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
\noindent \pmatrikel \quad \pname \\
\noindent \qmatrikel \quad \qname \\
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
}

\begin{document}
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\hwsol

\section*{Aufgabe 11.3}

\begin{enumerate}[1.]

    \item
        \begin{alignat*}{6}
            (2&x^3 && + 1&) / (3x^2 + x + 1) = \frac{2}{3}x - \frac{2}{9} \\
            -(2&x^3 + \frac{2}{3}&x^2 + \frac{2}{3}&x) \\ \cline{1-4}
            (& -\frac{2}{3}&x^2 - \frac{2}{3}&x + 1) \\
            -(& -\frac{2}{3}&x^2 - \frac{2}{9}&x - \frac{2}{9}) \\\cline{1-4}
            (-\frac{4}{9}&x + \frac{11}{9})
        \end{alignat*}

        Quotient: $\frac{2}{3}x - \frac{2}{9}$

        Rest: $- \frac{4}{9}x + \frac{11}{9}$

    \item
        \begin{align*}
            f(x) = 8x + 11 \\
            g(x) = 12x^2 + 9x + 11 \\
            f,g \in \mathbb{F}_{17}[x]
        \end{align*}

        \begin{alignat*}{3}
            (12&x^2 + 9&x + 11&) / (8x+11) = 10x + 15 \\
            -(12&x^2 + 8x &) \\ \cline{1-3}
                && x + 11& \\
            - (&& x + 12&) \\ \cline{1-3}
               && 16&
            \\
            \\
            \\
            8&x + 11& / (16) = 9x + 6 \\
            -(8&x) & \\ \cline{1-2}
               & 11& \\
            -(&11) \\ \cline{1-2}
            0
        \end{alignat*}

        $\Rightarrow$ $ggT(f,g) = 16 = 1 * (12x^2 + 9x + 11) - (8x+11)(10x+15)$

\end{enumerate}

\end{document}
