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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 3}
\\
{\bf April 20, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm}
Examination contains 8 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}
\begin{tabular}[t]{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
  1      &   10    &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  2      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  3      &   10    &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  8      &   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
Give three different ways to write the integer 252 as a product of prime
numbers.
\vspace*{4cm}

\item
Show that there are infinitely many prime numbers.
\vspace*{6cm}

\item
In $\Z[i]$ the following equation is true
\[
     5=(2+i)(2-i)=(1+2i)(1-2i)
\]
\begin{subproblem}
\item Show that both products are factorizations of $5$ into Gaussian primes.
         \vspace*{3cm}

\item How does the result of (a) relate to the fact that $\Z[i]$ is a
unique factorization domain?
          Give the exact relationship between the two factorizations.
          \vspace*{4cm}

\end{subproblem}

\newpage
\item
Factorize the following numbers into primes in $\Z[\sqrt{2}]$.
\begin{subproblem}
\item $4-5\sqrt{2}$,
          \vspace*{3cm}

\item $17$,
          \vspace*{3cm}

\item $87-58\sqrt{2}$,
          \vspace*{3cm}

\item $83$.
          \vspace*{3cm}
\end{subproblem}

\item
An $n$-cycle is a cycle of length $n$, i.~e. a permutation of the form
$(a_1\;a_2\;\ldots a_n)$ with
different $a_i$.
{\bf Prove:}
$A_n$ contains an $n$-cycle exactly when $n$ is odd.
\vspace*{6cm}

\item
Consider the symmetric group $S_6$.
\begin{subproblem}
\item Give three different elements of order $6$ in $S_6$.
          \vspace*{6cm}

\item Give a subgroup of $S_6$ with order 24. (You don't have to list all
elements of this subgroup!).
          \vspace*{6cm}

\item Give a subgroup of $S_6$ with order 5.
           \vspace*{6cm}

\end{subproblem}

\newpage
\item
Let $\sigma=(1\,5)(2\,4\,7\,3)$ and $\tau=(3\,5\,6\,2)(1\,4\,7)$ in $S_7$.
\begin{subproblem}
\item Calculate ${\rm sgn}\,\sigma$.
          \vspace*{3cm}

\item  Calculate $\sigma\tau^2\sigma^{-1}$.
           \vspace*{3cm}

\item  Find a permutation $\rho\in S_7$ so that
$\rho\sigma=(2\,5\,1\,4\,7\,3)$.
            \vspace*{4cm}
\end{subproblem}

\item
{\bf Prove by induction:}
$1\cdot2+2\cdot3+3\cdot4+\ldots+(n-1)\cdot n={\Ds \frac{(n-1)n(n+1)}{3}}$
for all $n\geq2$.
\\[1ex]
Give all details of the induction.
\vspace*{6cm}

\end{problems}
\end{document}
