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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}
%Import header
\hwsol

\section*{Aufgabe 11.1}

\begin{enumerate}[1.]

    \item Sei $p$ prim:

        \begin{align*}
            (a + b)^p &\equiv a^p + b^p &\mod p
        \end{align*}

        \begin{align*}
            (a + b)^p &\equiv \sum\limits^p_{k=0} \binom{p}{k} a^k b^{p-k} &\mod p \\
                      &\equiv a^p + b^p + \sum\limits^{p-1}_{k=1} \binom{p}{k} a^k b^{p-k} &\mod p \\
            *         &\equiv a^p + b^p &\mod p
        \end{align*}

        $*$: $\binom{p}{k} = p * \frac{(p-1)!}{k!(p-k)!}$

        $p$ prim $\Rightarrow$ $ggT(k!(p-k!), p) = 1$ $\Rightarrow$ $\frac{(p-1)!}{k!(p-k)!} \in \mathbb{Z}_p$

        $\Rightarrow$ $p | \binom{k}{p}$ für $1 \leq k \leq p-1$

        $\Rightarrow$ $\sum\limits_{k=1}^{p-1} \binom{p}{k} a^k b^{p-k} \equiv 0 \mod p$

        q.e.d.


    \item Sei $a,b \in \mathbb{N}, c = ggT(a,b)$

        \begin{align*}
            \varphi(a * b) &= \varphi(a) * \varphi(b) * \frac{c}{\varphi(c)}
        \end{align*}

        Zu zeigen: $\varphi(a) * \varphi(b) = \varphi(ggT(a,b)) * \varphi(kgV(a,b))$

        Seien $P'$ die gemeinsamen Primteiler von $a$ und $b$ und $A$ und $B$ die Mengen der disjunkten Primteiler von
        $a$ und $b$.

        \begin{align*}
            \varphi(a) * \varphi(b) &= \prod\limits_{p \in P'} p^{(a_p - 1)(b_p -1)} (p-1)^2 * \prod\limits_{p \in A}
            p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \\
                                    &= \prod\limits_{p \in P'} p^{(min(a_p,b_p)-1)(max(a_p,b_p)-1)} (p-1)^2 * \prod\limits_{p \in A}
            p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \\
                                    &= \prod\limits_{p \in P'} p^{min(a_p,b_p)-1} (p-q) * \left(
                                    \prod\limits_{p \in P'} p^{max(a_p,b_p)-1} (p-1) * \prod\limits_{p \in A}
            p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) \right) \\
                                    &= \varphi(ggT(a,b)) * \varphi(kgV(a,b)) \\
                                    \\
            \varphi(a * b) &= \varphi(a) * \varphi(b) * \frac{c}{\varphi(c)} \\
                           &= \varphi(ggT(a,b)) * \varphi(kgV(a,b)) * \frac{ggT(a,b)}{\varphi(ggT(a,b))} \\
                           &= \varphi(kgV(a,b)) * ggT(a,b) \\
                           &= \prod\limits_{p \in P'} p^{max(a_p,b_p)-1} (p-1) * \prod\limits_{p \in A}
                           p^{(a_p-1)} (p-1) * \prod\limits_{p \in B} p^{(b_p - 1)} (p-1) * \prod\limits_{p \in P'}
                           p^{min(a_p,b_p)} \\
                           &= \prod\limits_{p \in P'} p^{(a_p+b_p-1)} (p-1) * \prod\limits_{p \in A} p^{a_p-1} (p-1) *
                           \prod\limits_{p \in B} p^{b_p-1} (p-1) \\
                           &= \varphi(a*b)
        \end{align*}

        q.e.d.

\end{enumerate}


\end{document}
