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

\input{ExamMacros.tex}

\textheight9.0in
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\prof{Peter Cholak and Juan Migliore}
\class{Math 222 - Final}
\quizdate{Monday, May 7}
\pagestyle{empty}

\begin{document}

\heading



%\noindent Part I:  Definitions.\\

\begin{problem}(4 points each -- 20 points total) 
  {\bf Define} the following terms:\\
 
  \begin{subproblem}
    A {\em primitive element} of the Galois field $GF(p, P(x))$.
  \end{subproblem}
\vspace{1.5in}
\begin{subproblem}
The {\em index} of a subgroup $H$ of a finite group $G$.
  %The \emph{argument} of a complex number $z$. Must be a better defn.
\end{subproblem}\vspace{1.5in}
\begin{subproblem}
    A polynomial being \emph{solvable by radicals}.
  \end{subproblem}
\vspace{1.5in}
\begin{subproblem}
     A \emph{field}.  (You can assume the definition of a ring.)
  \end{subproblem}
\vspace{1.5in}
\begin{subproblem}
  The \emph{order} of an element $g$ in a group $G$.
  \end{subproblem}
\vspace{1.5in}
\end{problem}

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

\newpage

\begin{problem}(2 points each  -- 20 points total) 
  Answer {\bf True} or {\bf False} - no work
  required:\\
  \begin{subproblem}
    Every degree 3 polynomial over $\qq$ has a
  nonconstructible zero.
  \end{subproblem}
   \vspace{.6in}
   \begin{subproblem}
     There is a degree 5 polynomial over $\qq$ which is not solvable
     by radicals.
   \end{subproblem}
   \vspace{.6in}
   \begin{subproblem}
     The set of $2 \times 2$ matrices over the reals is an unital
     commutative ring.
   \end{subproblem}
 \vspace{.6in}
  \begin{subproblem}
    ${\mathbb Z}_6$ is a commutative ring but {\em not} an integral
domain.
  \end{subproblem}
  \vspace{.6in}
  \begin{subproblem}
   There exists a field with 32 elements.
  \end{subproblem}
\vspace{.6in}
  \begin{subproblem}
    If $G$ is a group and $H$ is a subgroup, then the identity element
$1_G$ is an element of every coset of $H$.
  \end{subproblem}
 \vspace{.6in}
  \begin{subproblem}
    $({\mathbb Z}_5,+)$ is a subgroup of $({\mathbb Z},+)$.
  \end{subproblem}\vspace{.6in}
  \begin{subproblem}
    The order of $(1 ~ 2 ~ 3 )$ is even in $S_3$.
  \end{subproblem}
\vspace{.6in}
  \begin{subproblem}
    Every group of order 5 is cyclic.
  \end{subproblem} 
  \vspace{.6in}
 
  \begin{subproblem}
    Let $H$ be a subgroup of the group $G$ and let $h$ be an element
  of $H$.  Then the group $<h>$ is contained in $H$.
  \end{subproblem} 

\end{problem}

\newpage

\begin{problem} (10 points)
  Find all the zeros of the polynomial $x^5(x^3 - 4) - (x^3 -4) $ in
  the complex numbers.  Which of these solutions are not
  constructible?  Why?
\end{problem}
\newpage
\begin{problem} (15 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)
 %% Here is a suggested problem; it might be too easy. 
%%Find all the primitive 6th roots of unity in $\zz_{7}$.
%\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 {\em all} the zeros of $f(x)$ in $\cc$.  
\end{problem}
\newpage
\begin{problem}(15 points)
  Construct the cyclic table for $GF(3,x^2+2x+2)$.
\end{problem}
\newpage
\begin{problem}(10 points)
  Does $S_8$ have a cyclic subgroup of order \  5? \ 11? \ 15?  
In each case, if so
  find such a subgroup.  If not, explain why not. 
\end{problem}
\newpage
\begin{problem}(10 points) 
{\bf (a).} Write down all the elements of $A_4$ {\em in disjoint cycle
notation}.
\vskip 4in

{\bf (b).}    Find all the cosets of $\{id, (1~3)(2~4)\}$ in $A_4$.  
\end{problem}
\newpage
\begin{problem}(10 points)
  Find \emph{all} subgroups of $D_3$.
\end{problem}
\newpage
\begin{problem}(10 points)
  Let $F = GF(p,P(x))$ where $P(x)$ is irreducible over $\Bbb{Z}_p$.
  Show that for any $a$ in $F$ there is at most one $p$th root of $a$.
\end{problem}
\newpage
\begin{problem}(10 points)
  Let $R$ be a commutative ring with unity and let $U(R)$ denote the
  set of units of $R$.  Show that $U(R)$ is a group under the
  multipliciation of $R$. (This group is called the group of units of
  $R$.)
\end{problem}
\newpage
\begin{problem}(10 points)
  Show that every finite cyclic group is abelian (i.e.\ is commutative)
\end{problem}

\end{document}