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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 1}
\\
{\bf February 9, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm}
Examination contains 7 problems on 6 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      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  2      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  3      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7      &   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{problem}
  Let $\alpha = -3 + 7i \in \Z[i]$. Find another number $\beta \in \Z[i]$
  so that $\alpha \beta = m (1+i)$ for some integer $m \in \Z$.
\end{problem}

\vspace{5cm}

\begin{problem}
  \begin{subproblem}
    Use the division algorithm for integers to find the base 5
    representation of the number $303$.  In other words, find
    coefficients
    $$0 \le r_i < 5$$
    so that
    $$
    303 = r_m 5^m + r_{m-1} 5^{m-1} + \cdots + r_1 5 + r_0.$$
  \end{subproblem}
\vspace{5cm}
  \begin{subproblem}
   What would you find a reasonable base 5 representation for $-1515$?
  \end{subproblem}
\end{problem}

\newpage

\begin{problem}
  Consider the ring $\Z[\sqrt{5}] = \{ a + b \sqrt{5} : a,b \in \Z\}$.

  \begin{subproblem}
    Find an integer $a$ so that the number $\alpha = a +\sqrt{5}$
    satisfies the inequality $0 < \alpha < 1$.
    Show also that $\alpha$ is a unit in $\Z[\sqrt{5}]$.
  \end{subproblem}
  \vspace{4cm}
  \begin{subproblem}
    Show that there exists a number $\beta \in \Z[\sqrt{5}]$ which lies
    between $1$ and $\dfrac{5}{4}$ on the number line.
  \end{subproblem}
  \vspace{4cm}
  \begin{subproblem}
    Assume that you know $\sqrt{5}$ to be an irrational number.
    Let $\alpha = a + b \sqrt{5} \in \Q(\sqrt{5})$, so $a,b \in \Q$.
Suppose $\alpha\not=0$.
    Find a formula for $c,d \in \Q$ so that
    $$\dfrac{1}{\alpha} = c + d \sqrt{5}.$$
    \vspace{3.5cm}

    Where do you need $\sqrt{5}$ being irrational? Explain this in detail.
  \end{subproblem}
\end{problem}

\newpage

\begin{problem}
  For each of the pairs $\alpha,\beta \in \Z[i]$ below, find
  $\gamma,\rho \in \Z[i]$ so that
  $$\beta=\alpha\gamma + \rho, \text{\quad and \quad   } N(\rho) < N(\alpha).$$

  \begin{subproblem}
    $\alpha = -1+3i$, $\beta= 21$
  \end{subproblem}
  \vspace{8cm}
  \begin{subproblem}
    $\alpha = 1-1i$, $\beta= 2 + 8i$
  \end{subproblem}
\end{problem}

\newpage

\begin{problem}
  For each pair of polynomials $f$ and $g$ below, find polynomials
  $q$ and $r$ so that
  $$g = qf + r \text{\quad and \quad} \text{deg}(r) < \text{deg}(f).$$
  In each case, regard the coefficients of $f$ and $g$ as coming
  from the coefficient ring $R$.
  \begin{subproblem}
    $R=\Z[\sqrt{2}]$, \quad $g= X^3 - X^2 + X + 1$, \quad $f = X - (1 +
\sqrt{2})$.
  \end{subproblem}
  \vspace{9cm}
  \begin{subproblem}
    $R = \Z_{13}$,\quad $g = X^4 + 12$, \quad$f =  X^3 + X^2 + X + 1$.
  \end{subproblem}
\end{problem}

\newpage
\begin{problem}
  Suppose that $y$ is a root of the polynomial
  $$X^5 + (n-1)$$
  in $\Z_n$, where $n$ is a positive integer. Show
  that $y$ is a unit in $\Z_n$.
\end{problem}
  \vspace{7cm}


\begin{problem}
  Let $a= 90$ and $b=168$.
  \begin{subproblem}
    Find $\gcd(a,b)$.
  \end{subproblem}
  \vspace{6cm}
  \begin{subproblem}
    Find $t,s \in \Z$ so that $ta + sb$ divides both $a$ and $b$.
  \end{subproblem}
  \vspace{6cm}


\end{problem}

\end{document}


