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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 13.3  \\ Spring 2023 \hfill Due April 19}}}

\large

\vspace{1cm}


\begin{exercise}
Let $p$ be an odd prime.  It can be shown that the multiplicative group $(\Z/p\Z)^{\times}$ is always cyclic.  Use this to prove that for any $a \in (\Z/p\Z)^{\times}$ one has
\[
\left( \frac{a}{p} \right) = a^{\frac{p-1}{2}},
\]
where $\left( \frac{\cdot}{p} \right)$ is the Legendre symbol.  This result is known as {\em Euler's criterion}.
\end{exercise}


\begin{exercise}
Let $\chi, \psi$ be characters of a finite abelian group $A$.  Define $(\chi \psi) : A \to \C^{\times}$ by $(\chi \psi)(a) = \chi(a) \psi(a)$ for all $a \in A$.  Show that $\chi \psi$ is a character of $A$.
\end{exercise}


\begin{exercise}
Let $A$ be a finite abelian group.  The {\em trivial character} of $A$ is defined by $\chi_0(a) = 1$ for all $a \in A$.  If $\chi$ is any character of $|A|$, prove that
\[
\sum_{a \in A} \chi(a) = \begin{cases} |A| & \text{ if } \chi = \chi_0, \\[5pt] 0 & \text{ otherwise.}\end{cases}
\]
[{\em Suggestion.}  Let $S$ denote the sum in question.  Show that for any $a \in A$ one has $\chi(a) S = S$.  This implies $S(\chi(a) - 1) = 0$.  Use this to conclude that $S=0$ if $\chi \ne \chi_0$.]
\end{exercise}


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