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\begin{document}
{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Mathematics 222.02: Algebraic Structures }
\\
{\bf Spring Semester 1998}
\\
{\bf Exam 2}
\\
{\bf March 6, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm}
Examination contains 6 problems on 5 sheets of paper
including the front cover.
Do all your work in this booklet and show your computations.
Calculators, books, and notes are not allowed.
\\[1.7ex]
\hline
\end{tabular}
\begin{center} {\bf Scores}
\\[1.5ex]
\begin{tabular}[t]{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
  1      &   15    &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  2      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  3      &   15    &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
Total    & 100\mbox{\  \ } & \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular}
\end{center}
\vspace*{1.4cm}
{\bf Sign the pledge:}
\\[1ex]
``On my honor, I have neither given nor received unauthorized aid on this
Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}
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\newpage
\begin{problems}
\item
\begin{subproblem}
\item  Find all units in $\Z_{21}$.
          \vspace*{4cm}
\item Calculate the quotient $\frac{15}{10}$ in $\Z_{21}$, that is, find
$15\cdot 10^{-1}$.
          \vspace*{6cm}
\end{subproblem}

\item
 Find all greatest common divisors of $11-10i$ and $-13i$ in $\Z[i]$ and give
 a Bezout-equation for one of them.
\vspace*{9cm}

\newpage
\item
Let $a$ and $b$ be relatively prime numbers.
Assume that there are numbers $x$ and $y$ so that
\[
        ax\equiv ay \text{ (mod $b$).}
\]
Show that it follows $x\equiv y$ (mod $b$).
\vspace*{9cm}

\item
\begin{subproblem}
\item Show that the polynomial $p(X)=2X^2+3X-7$ has no roots in $\Z$.
          \vspace*{5cm}

\item Show that $\sqrt[25]{2}$ is irrational.
          \vspace*{5cm}
\end{subproblem}


\newpage
\item
 Consider the ring $\Z_{41}$.
\begin{subproblem}
\item Assume that $a\not=1$ is an element of $\Z_{41}$  with $a^{15}=1$.
          Show that the order of $a$ is 5.
         \vspace*{2cm}
\item Find the order of 2. [Make sure that you get a divisor of 40.]
         \vspace*{3cm}
\item Find an element of order 5 in $\Z_{41}$.
          \vspace*{3cm}
\item Use the list you produced in (b) to find a square root of $-1$ in
$\Z_{41}$.
          \vspace*{3cm}
\item  How many square roots of 2 does $\Z_{41}$ have?  (You don't have to
find any!)
          \vspace*{2cm}
\item  Calculate the fraction $\frac{32}{10}$ in $\Z_{41}$, i.~e. find
$32\cdot 10^{-1}$.
          \vspace*{3cm}
\end{subproblem}

\item
\begin{subproblem}
\item Find a Bezout-equation for $a=8$ and $b=35$.
         \vspace*{7cm}

\item Use (a) to find a solution of the equation $8X\equiv 7$ (mod 35).
          \vspace*{6cm}
\end{subproblem}

\end{problems}
\end{document}
