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\newcommand{\Z}{\mathbb{Z}}
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\begin{document}
\prof{George McNinch}
\date{\today}
\examname{Exam 2}
\class{Math 222}
\heading

This examination contains 6 problems on 6 sheets of paper.  Show all
your work.  Calculators, books, and notes are not allowed.

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

\begin{center}
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  1      &   16     &       \\[.5pt] 
  2      &   16     &       \\[.5pt]
  3      &   17     &       \\[.5pt]
\end{tabular}
\quad
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  4      &   17     &       \\[.5pt]
  5      &   17     &        \\[.5pt]
  6      &   17     &        \\ \\
Total    & 100\mbox{\  \ } & \\
\end{tabular}
\end{center}



\begin{problem} Consider the ring $\Z_{20}.$
  \begin{subproblem}
    Find all the units in this ring.
  \end{subproblem}
      \vspace*{5cm} 
  \begin{subproblem}
    Find a number between 1 and 20 which represents the quotient
    $\frac{3}{11}$ in $\Z_{20}$.
  \end{subproblem}
 % \vspace{3cm}
%  \begin{subproblem}
%    Find all associates of 4 in $\Z_{20}.$
%  \end{subproblem}
\end{problem}

\newpage
\begin{problem}
  \begin{subproblem}
    Find all greatest common divisors of $1-i$ and $2-13i$ in $\Z[i]$
    and give a Bezout-equation for one of them.
  \end{subproblem}
  \vspace{8cm}
 % \begin{subproblem}
%    Use (a) to find $z \in \Z[i]$ so that $2 - z(1-i)$ is divisible by
%    $2-13i$
%  \end{subproblem}
  \begin{subproblem}
    Write $3=\sigma (1-i) + \tau (2-13i)$ for some choice of
    $\sigma,\tau \in \Z[i]$.
  \end{subproblem}
\end{problem}

\newpage


\begin{problem} Use the rational roots test to do the following problems.
  \begin{subproblem} Let $p$ be a prime number.
    Show that the polynomial $$f(X)=X^4+p^3X + p$$
    has no roots in
    $\Q$.
  \end{subproblem}
      \vspace{7cm}
  \begin{subproblem}
    Show that $\sqrt[98]{19}$ is irrational.  \vspace*{5cm}
  \end{subproblem}
\end{problem}

\newpage
\begin{problem}
  Let $p$ be a prime number, and suppose that $u$ is a unit of order 3
  in $\Z_p$.
  \begin{subproblem}
    Show that $p \equiv 1 \pmod 6$.
  \end{subproblem}
  \vspace{6cm}
  \begin{subproblem}
    Show that $u$ satisfies a monic polynomial equation of degree 3 of
    the form $X^3 - a = 0$ for some choice of $a \in \Z_p$. What is
    $a$?
  \end{subproblem}
  \vspace{6cm}
  \begin{subproblem}
    Use (b) to prove that there are precisely 2 units of order 3 in
    $\Z_p$. What are they?
  \end{subproblem}
\end{problem}

\newpage
\begin{problem}
  Consider the ring $\Z_{41}$.
%  \begin{subproblem}
%    Assume that $a\not=1$ is an element of $\Z_{41}$ with $a^{15}=1$.
%    Show that the order of $a$ is 5.    
%  \end{subproblem}
%  \vspace{4cm}
  \begin{subproblem}
    Assume you know that 5 has order 20 in $\Z_{41}$.  What is the
    order of $5^3$? Find $k \in \Z$ so that $5^k$ has order 5. 
  \end{subproblem}
  \vspace{8cm}
  \begin{subproblem}
    You are given that $13^2 \equiv 5 \pmod {41}$.  What is the order
    of 13 in $\Z_{41}$? (Be sure to justify your answer. Of course,
    you may use what you were told in part (a).)
  \end{subproblem}
%  \vspace{4cm}
%  \begin{subproblem}
%    Show that every unit in $\Z_{41}$ can be written as $13^k$ for
%    some $k \in \Z$. You should be able to do this without listing the
%    powers.
%  \end{subproblem}
  \vspace{4cm}
\end{problem}

\newpage
\begin{problem}
  Let $R$ be a commutative ring. Let $a \in R$ be a unit of order 14,
  let $b \in R$ be a unit of order 22, and let $c \in R$ be a unit of
  order 10.  Find the largest possible value of the order of $abc$.
  Give all details; don't just tell me the numerical answer. You might
  try first estimating the order of $ab$, and then of $(ab)c$.
\end{problem}
          
\end{document}

