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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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\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}
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\begin{document}
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\section*{Aufgabe 11.2}

Sei $p$ prim.

\begin{align*}
    f(x) = x^p - x \in \mathbb{F}_p[x]
\end{align*}

Nullstellen berechnen:

\begin{align*}
    x^p - 1 &\equiv 0 &\mod p \\
    x (x^{p-1}-1) &\equiv 0 &\mod p \\
    \Rightarrow (x \equiv 0 \mod p) \lor (x^{p-1} \equiv 1 \mod p)
\end{align*}

Gilt wegen Satz von Fermat $\forall x \in \{1,...,p-1\}$

$\Rightarrow$ $\forall x \in \mathbb{F}_p$ gilt: $x$ ist Nullstelle von $x^p - x$

\end{document}