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\begin{document}
\prof{George McNinch}
\class{Math 222}
\quizdate{\today}
\examname{Exam 1}

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}

\heading


This
Examination contains 7 problems on 6 sheets of paper
Do all your work in this booklet and show your computations.
Calculators, books, and notes are not allowed.


\noindent {\bf Points}
\\[1.5ex]
\begin{center}
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  1      &   10     &       \\[.5pt] 
  2      &   15     &       \\[.5pt]
  3      &   15     &       \\[.5pt]
  4      &   15     &       \\[.5pt]
\end{tabular}
\quad
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  5      &   15     &        \\[.5pt]
  6      &   15     &        \\[.5pt] 
  7      &   15     &        \\ \\
Total    & 100\mbox{\  \ } & \\
\end{tabular}
\end{center}

\vspace{3cm}
\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}

\newpage

\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{9cm}
  \begin{subproblem}
   What would you consider 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 $0 < \alpha < 1$.
  \end{subproblem}
  \vspace{5cm}
  \begin{subproblem}
    Show that there exists a number $\beta \in \Z[\sqrt{5}]$ which
    lies between $2$ and $\dfrac{9}{4}$ 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{4cm}
    
    Why do you need to know that $\sqrt{5}$ is irrational? Explain
    this in detail.
  \end{subproblem}
%  \vspace{4.5cm}
%  \begin{subproblem}
%    Find $c,d \in \Q$ so that
%    $$\dfrac{1 + \sqrt{5}}{3 + \sqrt{5}} = c + d \sqrt{5}.$$
%  \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{9cm}
  \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{6cm}

\begin{problem}
  Let $a= 90$ and $b=168$.
  \begin{subproblem}
    Find $\gcd(a,b)$.
  \end{subproblem}
  \vspace{4.5cm}
  \begin{subproblem}
    Find $t,s \in \Z$ so that $ta + sb$ divides both $a$ and $b$.
  \end{subproblem}
%  \vspace{4.5cm}
  %\begin{subproblem}
%   Which number $0 \le n < 90$ in $\Z_{90}$ coresponds to  $168$?
%  \end{subproblem}

\end{problem}

\end{document}



