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\prof{Chris Bendel and Peter Cholak} \class{Math 222 -  Exam 2}
\quizdate{Wednesday, April 14} \pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  Be sure to explain your
answers.\\

\begin{problem}(4 points each) {\bf Define} the following terms:\\
a) {\em irreducible} polynomial. 
\vskip1.7in
b) $GF(p,P(x))$ where $P(x)$ is an irreducible polynomial of degree $\nu$ over
$\mathbb{Z}_p$.
\vskip1.7in
c) {\em order} of an element in a group. 
\vskip1.7in
d) {\em primitive} element in a finite field with $p^{\nu}$ elements.\\
\end{problem}

\newpage

\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
  required:\\
  a) The polynomial $x^4 + 2x^2 + 1$ is irreducible over $\mathbb{Z}_3$.\vskip1.7in
  b) The order the number 2 in the group $(\mathbb{Z}_3,+)$ is 2.\vskip1.7in
  c) The set of 10th roots of unity in $\mathbb{C}$ is a group
  (under multiplication).\vskip1.7in
  d) Each element $\zeta \neq 0, 1$ in $GF(2,x^5 + x^2 + 1)$ is primitive.\\
\end{problem}

\newpage

\begin{problem} (10 points)
  Find the remainder when $x^{73}$ is divided by $x+2$ in
  $\mathbb{Z}_5$.
\end{problem}

\newpage

\begin{problem} (15 points)
  Consider the field $GF(2,x^4+x+1)$.  Let $\beta$ be the associated
  Galois imaginary.  $\beta$ is primitive. Why? Find the inverse of
  $\beta^2+\beta$ as a power of $\beta$.
\end{problem}
\newpage

\begin{problem} (10 points)
Let $\sigma =
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
5 & 6 & 8 & 1 & 4 & 2 & 9 & 3 & 7
\end{pmatrix}$.

a) Write $\sigma^{67}$ in disjoint cycle notation.

b) Is $\sigma^{35}$ even or odd?  Why?
\end{problem}

\newpage


\begin{problem} (10 points)
The set $G = \{4, 8, 16\}$ is a group under multiplication modulo 28.

a) Find the identity element of $G$.

b) Find the inverses of the remaining elements in $G$.
\end{problem}

\newpage

\begin{problem} (15 points)
  Let $p$ be a prime number and suppose that $\mathbb{Z}_p$ contains
  an element $c$ which is not a cube (in $\mathbb{Z}_p$).  Show that
  there exists a field with $p^3$-elements.
\end{problem}

\newpage

\begin{problem} (15 points)
  Let $G$ be a group.  Prove that if $x^2 = 1$ for each $x \in G$,
  then $G$ is abelian.
\end{problem}

\end{document}

