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\documentclass[12pt]{article}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{times,mathptm}
\input ExamMacros


\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}

\begin{document}
\prof{George McNinch}
\date{\today}
\examname{Final}
\class{Math 222}
\heading

This examination contains 5 problems (and one EXTRA CREDIT problem) on
6 sheets of paper.  Show all your work.  I doubt that calculators will
help much. Let me stress that ignoring the extra credit won't hurt
you! Your score will be regarded as being from 100 points.

\vspace{1cm}
\begin{centerline}
  {\bf \sc Points}
\end{centerline}

\begin{center}
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  1      &   20     &       \\[.5pt] 
  2      &   20     &       \\[.5pt]
  3      &   20     &       \\[.5pt]
\end{tabular}
\quad
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  4      &   20     &       \\[.5pt]
  5      &   20     &       \\[.5pt]
Extra    &   5     &       \\[.5pt]
Total    & 100\mbox{\  \ } & \\
\end{tabular}
\end{center}


\begin{problem}
  Let $p = 151$. It is a fact that $p$ is a prime number.  
    Show that each element of $\Z_{p}^\times$ has a 7th root. (Hint:
    consider the function $f: \Z_p^\times \to \Z_p^\times$ given by
    $f(x) = x^7$. Argue that $f$ is \emph{onto.})


 % \vspace{5cm}
%  \begin{subproblem}
%    Let $F = \Z_p[i]$. Since $p \equiv 3 \pmod 4$, we have seen in
%    class that $F$ is a field. Show that $F$ contains a unit of order
%    19, but that $\Z_p$ does not.
%  \end{subproblem}
\end{problem}

%\newpage
%\begin{problem}
%  Find a subgroup of order $7$ of the group $\Z_{49}^\times$, the
%  group of units of the finite ring $\Z_{49}$.  
%\end{problem}
\newpage
\begin{problem}
  For the following pairs of numbers, answer the given questions. Be
  sure to pay attention to which ring you are to consider the numbers
  in.
  \begin{subproblem}
    Find the greatest commond divisor of $a= 980, b = 3920 \in \Z.$
  \end{subproblem}
  \vspace{7cm}
  \begin{subproblem}
    Let $\alpha = 5 + 5i, \beta = 7 - i \in \Z[i].$ Factor $\alpha$
    and $\beta$ into primes in $\Z[i]$, and then find their greatest
    common divisor.
  \end{subproblem}
\end{problem}

\newpage
\begin{problem}
  Let $\alpha = a + b \sqrt{-3} \in \Q(\sqrt{-3})$, where $a,b \in \Q$
  and put
  $$M_\alpha = 
  \begin{bmatrix}
    a & -3b \\
    b & a \\
  \end{bmatrix}.$$
  Let $$F_\alpha(T) = \det \left (
    \begin{bmatrix}
      T & 0 \\
      0 & T \\
    \end{bmatrix}
    -
    M_\alpha \right )$$
    denote the \emph{characteristic polynomial} of $M_\alpha$.
    \begin{subproblem}
      Show that $\alpha$ and $\overline{\alpha}$ are roots of
      $F_\alpha(T)$. (Hint: Use the formula $F_\alpha(\alpha) = \det
      \left (
        \begin{bmatrix}
          \alpha & 0 \\
          0 & \alpha \\
        \end{bmatrix}
        -
        M_\alpha
      \right )$ and a similar formula for $F_\alpha(\overline{\alpha})$.)
    \end{subproblem}
  
\vspace{5cm}
    \begin{subproblem}
      If $\alpha + \overline{\alpha} \in \Z$ and $N(\alpha) \in \Z$,
      show that the polynomial $F_\alpha(T)$ has \emph{integer
        coefficients.} (Thus, $\alpha$ and $\overline{\alpha}$ are
      \emph{algebraic integers.})
    \end{subproblem}

\vspace{5cm}
    \begin{subproblem}
      Let $\alpha = \dfrac{1 + \sqrt{-3}}{2} \in \Q(\sqrt{-3})$. Use
      (b) to show that $\alpha$ is an algebraic integer (despite the
      denominator).
    \end{subproblem}
\end{problem}


\newpage

%\begin{problem}
%  Let $u$ generate the group of units of $\Z_{19}$. Find a unit $v \in
%  \Z_{19}$ of order 3 (you may describe $v$ in terms of $u$). 

%  \vspace{4cm}
%  Find all
%  roots in $\Z_{19}$ of the polynomial
%  $$f(T) = T^3 - 7.$$
%\end{problem}

\begin{problem}
  Let $G = D_6$ be the group of symmetries of a regular hexagon.
  Label the vertices of the hexagon with the numbers $1,2,\dots,6$,
  and regard the elements of the group $G$ as permutations of these
  6 vertices.

  Let $H$ denote the set of all elements $\sigma \in G$ so that
  $\sigma$ permutes the vertices $S=\{1,3,5\}$.  More formally,
  $$H = \{ \sigma \in G : \sigma(1) \in S,\ \sigma(3) \in S, \text{
    and }\sigma(5) \in S\}.$$

  \begin{subproblem}
    Show that $H$ is a subgroup.
  \end{subproblem}
\vspace{5cm}
  \begin{subproblem}
    Find the order $|H|$ of $H$.
  \end{subproblem}
\vspace{5cm}
  \begin{subproblem}
    Let $\tau = (14)(23)(56) \in G$. Show that each element $\gamma$
    in the coset $\tau H$ satisfies $\gamma(1) \in \{2,4,6\}$.
\end{subproblem}
\end{problem}

%\begin{problem} 
%\label{problem:omega}
%This problem still concerns $\Z[\sqrt{3}]$.
%\begin{subproblem}
%  Let $\omega = 2 + \sqrt{3}$. Find $\dfrac{1}{\omega}$. Show that
%  $\omega$ is a unit in $\Z[\sqrt{3}]$. (i.e. $\omega \in
%  \Z[\sqrt{3}]^\times$, the group of units.) 
%\end{subproblem}
%\vspace{8cm}
%\begin{subproblem}
%  Show that $\omega$ generates an
%  infinite cyclic subgroup of $\Z[\sqrt{3}]^\times$.
%\end{subproblem}
%\end{problem}

\newpage
\begin{problem} Let $S_n$ denote the symmetric group on the set $\{1,2,\dots,n\}$,
  and let $A_n$ denote the alternating group (the subgroup of even
  permutations).
  \begin{subproblem}
    Show for $n>3$ that $A_n$ contains a permutation of order 2. (Let
    me remind you that \emph{transpositions} are odd permutations and
    hence aren't in $A_n$.)
\end{subproblem}
\vspace{5cm}
\begin{subproblem}
  Is the conclusion of (a) true when $n=3$? Hint: what is $|A_3|$?
\end{subproblem}
\vspace{5cm}
\begin{subproblem}
  Show for $n>3$ that $A_n$ is non-abelian (i.e. not commutative.)
\end{subproblem}
\end{problem}
\newpage
\noindent \textbf{EXTRA CREDIT:} (5 points)
Let $\omega =2 + \sqrt{3}\in \Z[\sqrt{3}]$. Write
$$\omega^k = a(k) + b(k) \sqrt{3}$$
for $k \ge 0$, where $a(k), b(k) \in \Z$.
Show that 
$$\lim_{k \to \infty} \dfrac{a(k)}{b(k)} = \sqrt{3}.$$
\end{document}

