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\prof{Chris Bendel and Peter Cholak} \class{Math 222 - Quiz 7}
\quizdate{Wednesday, March 17} \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}
  Problem 4 from Section 6.3.
\end{problem}


\begin{problem}
  Consider the Galois Field $F \equiv \operatorname{GF}(3,x^2 + x + 2)$.  Let
$\alpha$ be the associated Galois imaginary.

(a) Compute all powers of $\alpha$, i.e.\ $\alpha$, $\alpha^2$, $\alpha^3$,
\dots.  What is the least power for which $\alpha^k = 1$?

(b) Find the inverse of each nonzero element in $F$.  {\em Hint:} Use part
(a).

(c) By definition $\alpha$ is one solution to $x^2 + x + 2 = 0$ over
$\mathbb{Z}_3$.  There should of course be another solution.  It is also an
element of $F$.  Find this element.  Is it a power of $\alpha$?  {\em Hint:}
Use long division.
\end{problem}

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