%%  answers for 1a, 3.
%% problem and answer for 2d and 2e,
%% approve change in points for proof problems.

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\prof{Peter Cholak and Juan Migliore} \class{Math 222 - Exam 2}
\quizdate{Wednesday, April 25} \pagestyle{empty}

\begin{document}

\heading

\noindent {\bf Except where noted, be sure to show all your work.}

\bigskip

\begin{problem}(4 points each for a total of 20 points) 
  {\bf Define} the following terms: 
  
  a) \emph{irreducible polynomial}. \vspace{1.2in}

  b) \emph{$GF(p,P(x))$}, where $P(x)$ is irreducible and $p$ is
  prime. \vspace{1.4in}
  
  c) an \emph{even permutation}.  \vspace{1.2in}
   
 d) a \emph{group}.
  \vspace{1.4in}
  
 e) The \emph{symmetric group $S_n$}.
  
\end{problem}

\newpage
  
\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
  required:

 a) The dihedral group $D_4$ is not abelian. \vspace{.8in}
    
  b) The order of $(\mathbb{Z}_n, +)$ is $n$. \vspace{.8in}
    
  c) The set of \emph{primitive} $8$th roots of unity in
  $\mathbb{Z}_{17}$ is a group (under multiplication). \vspace{.8in}
   
  d)  Let $p$ be a prime number.  The multiplicative group $({\mathbb
Z}_p \backslash \{ 0\}, \cdot)$ has at least one element of order
$p$.  (Notice this says $p$ and not $p-1$.) 

\vspace{.8in}

  e)   Let $p$ be a prime number and let $P(x)$ be a polynomial of
degree $d$ that is irreducible over $\mathbb{Z}_p$.  Let $G = (GF(p,P(x))
\backslash \{0\}, \cdot)$, the multiplicative group of $GF(p,P(x))$. 
Then there must exist an element $x$ of $G$ satisfying $|G| = o(x)$.
\end{problem}

\vspace{.8in}

%\newpage

\begin{problem} (15 points)
  Work over $\mathbb{Z}_5$.  For each $a \in \mathbb{Z}_5$, state
  whether $x^2-a$ is irreducible and if not factor $x^2-a$ into
  irreducible factors.
\end{problem}

\newpage

\begin{problem} (10 points)
 Consider the Galois field $F = GF(11,x-1)$.

\begin{itemize}
\item[a.] How many elements does $F$ have?

\vspace{.5in}

\item[b.] Find {\em one} primitive element of $F$.

\vspace{1in}

\item[c.] In terms of powers of the element you found in part (b), find
{\em all } primitive elements of $F$.
\end{itemize}
\end{problem}

\vspace{1.5in}

\begin{problem}
(10 points)
Consider the permutation
\[
\sigma =
\left (
\begin{array}{cccccccc}
1 & 2 & 3 & 4 & 5 \\
3 & 5 & 1 & 4 & 2 
\end{array}
\right )
\]
\begin{itemize}
\item[a.] Write $\sigma$ as a product of disjoint cycles.

\vspace{.5in}

\item[b.] Write $\sigma$ as a product of 3-cycles.

\vspace{.8in}

\item[c.] What is $\sigma^{1234567}$?
\end{itemize}
\end{problem}

\newpage

\begin{problem}
(15 points)
Consider the dihedral group $D_3$ (the symmetries of an equilateral
triangle).  

\begin{itemize}
\item[a.] What is the order of $D_3$?

\vspace{.8in}

\item[b.] Let $R$ denote clockwise rotation of 120 degrees and let $F$
denote a flip about the vertical axis.  Describe all the elements of
$D_3$ and write them down in terms of $R$, $F$ and the identity.

\vspace{2.5in}

\item[c.] For each integer $k$ between 1 and 6 (inclusive) list the
elements of $D_3$ of order $k$, or else state that no such element
exists.
\[
\begin{array}{l}
\hbox{order 1:} \\
\hbox{order 2:} \\
\hbox{order 3:} \\
\hbox{order 4:} \\
\hbox{order 5:} \\
\hbox{order 6:}
\end{array}
\]
\end{itemize}
\end{problem}

\newpage

\begin{problem} (10 points)
  Let $a$ and $b$ be two elements of a field.  Prove that $a\cdot b =
  0$ in $F$ if and only if $a$ or $b$ is zero.
\end{problem}

\vspace{3in}


\begin{problem} (10 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 exactly $p^3$ elements.
\end{problem}


\end{document}



