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\prof{Chris Bendel and Peter Cholak} \class{Math 222}
\quizdate{Wednesday, February 10} \pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  It is suggested that you
first write out a draft of your proposed questions and then carefully
rewrite that draft to get your final version.  You do \emph{not} have
to write the answers on this sheet of paper.\\

\begin{problem}
Let $p$ be a prime number.  Show that every $p$th root of unity,
except 1, is primitive.
\end{problem}

\begin{problem}
Consider the set of integers modulo 8, $\mathbb{Z}_8$.\\  (a) Write
out the addition and multiplication tables for $\mathbb{Z}_8$. \\ (b)
Identify the additive inverse of each element in $\mathbb{Z}_8$.\\ (c)
Identify those elements which have a multiplicative inverse and what
their inverses are.\\  (d) What is the sum $1 + 2 + 3 + 4 + 5 + 6 + 7$
modulo 8?
\end{problem}

\begin{problem}
Let $n$ be a positive integer and consider the sum $1 + 2 + 3 + \dots + n-1$
modulo $n$.  Show that if $n$ is odd, this sum is zero in $\mathbb{Z}_n$ and
if $n$ is even then the sum is $n/2$ in $\mathbb{Z}_n$.  {\em Hint:} Use
the formula for this sum which we saw when discussing induction.
\end{problem}

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