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\class{Math 222 -- Section 02}
\quizdate{\today}
\examname{Practice Exam 1}

\begin{document}
\heading

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

\begin{problem}
  Let $\alpha = 5 + 6i \in \Z[i]$. Find another number $\beta \in \Z[i]$
  so that $\alpha \beta = m i$ for some integer $m$. (i.e. $m \in \Z$).
\end{problem}

\begin{problem}
  Let $\alpha = a + bi \in \Z[i]$ with $0 \le a < b$.  Show that
  $\alpha^2$ has negative real part. In other words, if 
  $$\alpha^2 = c + di \quad c,d \in \Z$$
  then show that $c < 0$.

  Can you give a geometric explanation of this? Characterize the angle
  that the vector $\alpha$ makes with the $x$-axis.
\end{problem}

\begin{problem}
  \begin{subproblem}
    Show that the equation $X^4 + 1 = 0$ has a solution in $\Z_{17}$.
  \end{subproblem}
  \begin{subproblem}
    Find four solutions to $X^4 +1 = 0 $ in $\Z_{17}$.
  \end{subproblem}
  \begin{subproblem}
    Regarding coefficients as coming from $\Z_{17}$, factor the
    polynomial $X^4 + 1$ as a product of degree 1 terms.
  \end{subproblem}
  
\end{problem}

\begin{problem}{} 
  Recall that a non-zero element $b$ of a ring $R$ is a \emph{zero
    divisor} if there is another non-zero element $c$ of the ring
  so that $bc = 0$. An element $b$ of a ring $R$ is a \emph{unit}
  provided there is another element $d$ of the ring so that
  $bd = 1$.
  \begin{subproblem}
    \textbf{Prove:} If $b$ is an element of a ring $R$, and $b$ is a
    zero divisor, then $b$ is not a unit in $R$.
  \end{subproblem}
  \begin{subproblem}
    Show that the number $4$ regarded as an element of $\Z_{28}$ is not
    a unit.
  \end{subproblem}
  \begin{subproblem}
    Find a unit $u$ different from $1$ or $-1=27$ in $\Z_{28}$.
  \end{subproblem}
\end{problem}

\begin{problem}
  \textbf{Prove} that $\sqrt{3}$ is an irrational number.
\end{problem}

\begin{problem}
  The rings $\Z[\sqrt{3}]$ is constructed in exactly the same manner
  as $\Z[\sqrt{2}]$. Answer the following questions concerning this ring.
  \begin{subproblem}
    Suppose that $a + b \sqrt{3}$ and $c + d \sqrt{3}$ are elements
    of $\Z[\sqrt{3}]$; thus $a,b,c,d \in \Z$.
    If $$a + b\sqrt{3} = c + d\sqrt{3},$$ show that $a=c$ and $b=d$.
  \end{subproblem}
  \begin{subproblem}
    Estimate the size of $\sqrt{3} -1$. Show that the ring
    $\Z[\sqrt{3}]$ contains an infinite sequence of numbers between
    $0$ and $1$ which approaches $0$.
  \end{subproblem}
  \begin{subproblem}
    Find a number in $\Z[\sqrt{3}]$ between $1$ and $\dfrac{5}{4}$.
  \end{subproblem}
\end{problem}

\begin{problem}
  Find 3 units in $\Z[\sqrt{3}]$ other than $1$ and $-1$.
\end{problem}

\begin{problem}
  Use the division algorithm for integers to find the base 7
  representation of the number $2002$.  In other words, find
  coefficients
$$0 \le r_i < 7$$
so that
$$ 2002 = r_m 7^m + r_{m-1} 7^{m-1} + \cdots + r_1 7 + r_0.$$
Do the same thing for $14014$.
\end{problem}

\begin{problem}
  For each of the pairs $\alpha,\beta \in \Z[i]$ below, find 
  $\gamma,\rho \in \Z[i]$ so that
  $$\beta=\alpha\gamma + \rho,$$
  and so that
  $$N(\rho) < N(\alpha).$$
  \begin{subproblem}
    $\alpha = 1+2i$, $\beta= -5 + 10i$
  \end{subproblem}
  \begin{subproblem}
    $\alpha = 4-3i$, $\beta= 12$
  \end{subproblem}
\end{problem}

\begin{problem}
  \begin{subproblem}
    Let $\alpha = a + b \sqrt{3} \in \Q(\sqrt{3})$, where $a,b \in \Q$ are
    rational numbers, and suppose that $\alpha \not = 0$.
    Give a formula for $c,d \in\Q$ so that
    $$\dfrac{1}{\alpha} = c + d\sqrt{3}.$$
    Convince me that you aren't dividing by $0$!
  \end{subproblem}
  \begin{subproblem}
    Compute the quotient $\dfrac{1 + \sqrt{3}}{2 + 3\sqrt{3}}.$
  \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}
    $R=\Q$, \quad $g= \dfrac{1}{12}x^4 + x^3  + 2x + 1$, \quad $f =
    \dfrac{d^2g}{dx^2}$ (the second derivative of $g$).
  \end{subproblem}
  \begin{subproblem}
    $R = \Z_{11}$,\quad $g = x^5 +10$, \quad$f = x^4 + x^3 + x^2 + x + 1$.
  \end{subproblem}
  \begin{subproblem}
    $R = \Z_{11}$, \quad $g = x^4 + x^3 + x^2 + x + 1$, \quad$f = x+7$.
  \end{subproblem}
\end{problem}

\begin{problem}
  Consider the polynomial $g(x) = x^3 -3x^2 + 4x -2$. Show that
  the Gaussian integer $\alpha = 1+i$ is a root of $g(x)$. Compute
  the quotient $g(x)/f(x)$ where 
  $f(x) = x-\alpha=x - (1+i)$; the resulting polynomial should have coefficients
  in $\Z[i]$.
\end{problem}

\begin{problem}
  Let $a = 1260$ and $b=1650$.
  Find $\gcd(a,b)$.
  Find integers $s$ and $t$ so that $sa + tb = \gcd(a,b)$.
\end{problem}

\end{document}

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