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\begin{flushright}
  Your Name Goes Here
\end{flushright}


\begin{center}
  \textbf{Math 25}\\ Homework 8
\end{center}

\begin{enumerate}
\item Determine congruence conditions which characterize all odd
  primes $p$ for which $11 \in~Q_p$.

\item Show that 3 is a primitive root of every Fermat number which is
  a prime greater than 3.

\item Let $p$ be an odd prime, $a, b \in \Z$ with $p\nmid a$.  Show
  that
  \[\sum_{x=0}^{p-1}\Legendre{ax+b}{p} = 0.
  \]

\item Let $\mu(n)$ be the M\"obius function defined for $n\ge 1$ by:
  \[\mu(n) =
    \begin{cases}
      1&n=1\\0&\mbox{if $n$ is not squarefree}\\
      (-1)^t&\mbox{if $n = p_1\cdots p_t$ is the product of $t$
        distinct primes.}
    \end{cases}
  \]

  Your book defines $\mu(n)$ differently for which the definition
  above is Theorem 8.8.  Let's prove that their definition follows
  from ours, that is: Prove that for all integers $n \ge 1$
  \[\sum_{d\mid n} \mu(d) =
    \begin{cases}
      1& \mbox{if $n=1$}\\
      0&\mbox{if $n > 1$}.
    \end{cases}\] Hint: It might be useful first to show that $\mu$ is
  a multiplicative function.
        
        

\item Show that for all integers $n \ge 1$
  \[\frac{n}{\phi(n)} = \sum_{d\mid n} \frac{\mu^2(d)}{\phi(d)},
  \]
  where $\mu$ is the M\"obius function defined above.
\end{enumerate}
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