\documentclass[12pt]{article}
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%\usepackage{times,mathptm}
\usepackage{amsmath,amsfonts}

\def \zz {\mathbb{Z}}
\def \cc {\mathbb{C}}
\def \rr {\mathbb{R}}
\def \qq {\mathbb{Q}}

\input{ExamMacros.tex}

\textheight9.0in
\topmargin-0.3in

\prof{Chris Bendel and Peter Cholak}
\class{Math 222 - Final}
\quizdate{Thursday, May 6}
\pagestyle{empty}

\begin{document}

\heading


%\noindent Part I:  Definitions.\\

\begin{problem}(4 points each) {\bf Define} the following terms:\\
 
  a) A \emph{subgroup}. \\

\vspace{1.8in}

  b) The \emph{modulus} of a complex number $z$. \\

\vspace{1.8in}

  c) A polynomial being \emph{solvable by radicals}.\\

\vspace{1.8in}

  d) An \emph{odd permutation}.\\

\vspace{1.8in}

  e)  A \emph{cyclic} group.\\

\vspace{1.8in}

  f) An {\em irreducible} polynomial.\\

\vspace{1.8in}
\end{problem}

%\noindent Part II:  True/False.\\

\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
  required:\\
    
  a) Every degree 3 polynomial over $\qq$ has a
  nonconstructible zero.\\

  \vspace{1.8in}

  b) There is a degree 5 polynomial over $\qq$ which is not
  solvable by radicals.\\
  
  \vspace{1.8in}

  c) Let $H$ be a subgroup of the group $G$ and let $h$ be an element
  of $H$.  Then the group $<h>$ is contained in $H$. \\
  
 \vspace{1.8in}

 d) The order of $(1 ~ 2 ~ 3 )$ is even in $S_3$. \\
  
 \vspace{1.8in}

e) The quaternions are isomorphic to a subgroup of $(\zz_{16},+)$.\\
  
  \vspace{1.8in}

  f) Let $f: (\cc,+) \to (\cc,+)$ by $f(a + bi) = a - bi$.  $f$ is 
  an isomorphism.\\

\end{problem}
\newpage
%\noindent Part III: Computational Problems.  \\

\begin{problem} (10 points)
  Find all the zeros of $x^5 + 32$ in the complex numbers.
\end{problem}
\newpage
\begin{problem} (10 points)
  Find all the primitive 6th roots of unity in $\zz_{7}$.
\end{problem}
\newpage
\begin{problem} (10 points)
  Does $S_8$ have a cyclic subgroup of order $3,5,7,11,13,15$?  If so
  find such a subgroup.  
\end{problem}
\newpage
\begin{problem} (10 points)
  Are $D_6$ and $A_4$ isomorphic groups? Why or why not? %(Hint:
  %consider the order of the elements in these groups.)
\end{problem}
\newpage
\begin{problem} (10 points)
  Use the Euclidean algorithm to find $d = \gcd(304,399)$ and then
  find integers $x$ and $y$ such that $d = 304x + 399y$.
\end{problem}
\newpage
\begin{problem} (10 points)
  Construct the cyclic table for $GF(3,x^2+2x+2)$.
\end{problem}
\newpage
\begin{problem} (10 points)
  Given that $3i$ is a zero of $f(x) = x^3 - 6ix^2 - 11x +6i$ over
  $\cc$.  Find all the zeros of $f(x)$ in $\cc$.  
\end{problem}
\newpage
%\noindent Part IV: Proofs.\\

\begin{problem} (15 points)
  Let $p$ be a prime number and $P(x)$ be an irreducible polynomial of
  degree $\nu$ over $\mathbb{Z}_p$.  Suppose that $n$ is a positive
  integer relatively prime to $p^{\nu} - 1$.  Prove that there is
  exactly one $n$th root of unity in $GF(p,P(x))$.
\end{problem}

%\newpage
%\begin{problem} (15 points)
%  Let $f: G \to H$ be an isomorphism of groups.  Prove that $f(1_G) =
%  1_H$.
%\end{problem}

\newpage
\begin{problem} (15 points)
  Let $f$ be a function from a group $(G,*)$ to a group $(H,*)$ such
  that $f(a*b)=f(a)*f(b)$.  The kernel of $f$ is $\{g \in G : f(g) =
  1_H\}$.  Show the kernel of $f$ is a subgroup of $G$. 
\end{problem}

\newpage
\begin{problem} (15 points)
  Let $p$ be a prime number and $G$ be a finite group of order $p$.

  a) Show that $G$ is cyclic.

  b) Use part (a) to show that there is only one group of order $p$ up
  to isomorphism.
\end{problem}

\end{document}