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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 Final Exam}
\\
{\bf May 6, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm}
Examination contains 14 problems on 7 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      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   \mbox{\ } 5     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7       &  15    &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular} \qquad
\begin{tabular}[t]{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
  8        &   \mbox{\ } 5     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  9        &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 10       &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 11       &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  12      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  13       &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  14        &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
Total    & 150\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\begin{problems}
\item
Find the set of all Gaussian integers which are multiples of $i$.
\vspace*{3cm}

\item
\begin{subproblem}
\item Give all roots of the polynomial $f=X^2+X+2$ in $\Z_6$.
         \vspace*{5cm}

\item Give all roots of the polynomial $f=X^2+4X+3$ in $\Z_8$.
          Factorize $f$ over $\Z_8$ as much as possible. [Check your answer!]
          \vspace*{6cm}

\end{subproblem}

\junk{
\item
Give (in an efficient way) 5 elements   of $\Z[\sqrt{2}]$ with norm 4 and
which are between $0$
 and $0.25$.
\vspace*{5cm}
}
\item
Consider the additive group $\Z_{18}$. Give all different cosets of the
subgroup generated by
the element $3$ in $\Z_{18}$.

\newpage
\item
Given the polynomials $f=2X^4+10X^3+2X+5$ and $g=X^2+3X+4$ with
coefficients in $\Z_{12}$.
Find polynomials $a$ and $r$ with coefficients in $\Z_{12}$ so that
$f=ag+r$ and $\deg r<\deg g$.
\vspace*{8cm}


\item
Find all roots of the polynomial $p=X^4+4X^3+6X^2+5X+2$ in $\Q$.
\vspace*{7cm}

\item
Find ${\Ds \frac{6}{7}}$ in $\Z_{12}$. Express the result as a number
between 0 and 11.
\vspace*{5cm}

\newpage
\item
Consider the multiplicative group $\Z_{61}\backslash\{0\}$.
\begin{subproblem}
\item Show that this group contains no element of order 9.
          \vspace*{5cm}

\item Calculate the order of $3$.
           \vspace*{5cm}

\item Give an element of order 5.
          \vspace*{5cm}

\end{subproblem}

\item
Find a solution of the equation $7X\equiv 3$ (mod 12).
\vspace*{5cm}

\item
\begin{subproblem}
\item Define carefully the notion of a zero divisor.
          \vspace*{4cm}

\item Let $a$ and $n$ be nonzero integers. Show: If $a$ and $n$ are not
relatively prime, then $a$ is a zero divisor in $\Z_n$.
         \vspace*{6cm}

\end{subproblem}

\item
Prove by induction:
\[
              1^3+2^3+\ldots +n^3=\Bigg(\frac{n(n+1)}{2}\Bigg)^2 \qquad
\mbox{\rm for all }n\geq1
\]
\vspace*{7cm}

\junk{
\item
Factorize into primes in $\Z[\sqrt{2}]$.
\begin{subproblem}
\item $49$,
\item $-3+8\sqrt{2}$,
\item $30-20\sqrt{2}$.
\end{subproblem}
}

\item
\begin{subproblem}
\item Factorize $29$ into Gaussian primes.
          \vspace*{3cm}

\item Use a) to find a greatest common divisor of $29$ and $1+12i$ in $\Z[i]$.
         \vspace*{7cm}

\end{subproblem}

\item
Show that if $\sigma,\,\tau\in S_n$, then $\sigma\tau^2\sigma\in A_n$.
\vspace*{4cm}

\item
Show that if $G$ is a group of prime order $p=|G|$, then $G$ is a cyclic
group.
\vspace*{5cm}

\newpage
\item
\junk{
\begin{minipage}[t]{12cm}
}
Consider the group $D_4$ of all symmetries of the square.
Remember $D_4$ is a subgroup of $S_4$.

\begin{subproblem}
\item
\begin{minipage}[t]{13cm}
List all elements of $D_4\cap A_4$; use the picture for the notation of the
vertices.
\end{minipage}
\hfill\begin{minipage}[t]{4cm}
\unitlength.5cm
\hspace*{3.5cm}\begin{picture}(-5,-5)
\put(0,0){\line(0,-1){4}}
\put(0,0){\line(-1,0){4}}
\put(-4,0){\line(0,-1){4}}
\put(0,-4){\line(-1,0){4}}
\put(.3,.3){$2$}
\put(-4.6,.3){$1$}
\put(.3,-4.7){$3$}
\put(-4.6,-4.7){$4$}
\end{picture}
\end{minipage}
\vspace*{4cm}

\item Find the smallest subgroup of $D_4$ which contains $(1\,2\,3\,4)$ and
$(1\,4)(2\,3)$.
            \vspace*{4cm}

\junk{
\item Which of the following elements are in $D_4$
          (use the above notation for the vertices of the square).
          \\[1ex]
          (i)\ \ $(1\,3)$ \hspace*{1.5cm} (ii)\ \
$(1\,3)(2\,4)$\hspace*{1.5cm} (iii)\ \ $(1\,2)$.
          \vspace*{3cm}
}
\item Let $\sigma=(1\,3\,2)$ and $\tau=(2\,4\,3\,1)$ be elements of $S_4$.
          Find
          \\[1ex]
           (i)  $(\sigma\tau)^{-1}$,
             \vspace*{2.3cm}

            (ii) sgn\,$(\sigma\tau^2)$,
               \vspace*{2.3cm}

            (iii) $\rho\in S_4$ so that $\sigma\rho\tau=1$,
              \vspace*{2.3cm}

             (iv) the order of $\tau^2$.
               \vspace*{2.3cm}

\end{subproblem}

\end{problems}
\end{document}
