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\prof{Chris Bendel and Peter Cholak} \class{Math 222 - Exam 1}
\quizdate{Wednesday, February 24} \pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  Be sure to explain your
answers.\\

\begin{problem}(4 points each) {\bf Define} the following terms:

a) $\sqrt[n]{1}$ \quad (over the complex numbers for a positive integer $n$).

\vskip1.7in
b) A {\em constructible}  number.

\vskip1.7in
c) A {\em primitive} $n$th root of unity for a positive integer $n$.

\vskip1.7in
d) The {\em multiplicative inverse} of a number $a$ in $\mathbb{Z}_n$ ($n$
a positive
integer).
\end{problem}

\newpage
\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
required:

a) If $\operatorname{arg}(z) = \phi$, then $\operatorname{arg}(z^2) = \phi^2$.

\vskip1.5in
\ans

b) $|3 + 4i| = 5$.

\vskip1.5in
\ans

c) $5\cdot6\equiv 5 \quad (\operatorname{mod} 7).$

\vskip1.5in
\ans

d) The coefficient of $x^8y^2$ in $(x+y)^{10}$ is 45.

\vskip1.4in
\ans
\end{problem}

\newpage
\begin{problem}(15 points) Find all solutions to $x^6-8=0$.  Explain
briefly why these
  solutions are constructible?  (You do not need to simplify your answer to the
   form $a + bi$.)
\end{problem}

\newpage
\begin{problem}(15 points) Find the multiplicative inverse of 56 in
$\mathbb{Z}_{67}$.
\end{problem}

\newpage
\begin{problem}(15 points) Prove that $n^5 - n$ is divisible by $30$ for
any positive integer
$n$.
\end{problem}

\newpage
\begin{problem}(15 points)  Let $n$ be a positive integer and $\zeta$ be
the first $n$th root
of unity (i.e.\ $\zeta \neq 1$) and let $\alpha = \zeta^k$ be any other
$n$th root of unity.
Prove that there exists
  an integer $m$ such that $\zeta = \alpha^m$ if and only if $k$ has a
  multiplicative inverse in $\mathbb{Z}_{n}$.
\end{problem}

\newpage
\begin{problem}(15 points) Let $a$ and $b$ be nonzero integers such that $g
= (a,b)$.  Prove
  that $\displaystyle\left(\frac{a}{g},\frac{b}{g}\right)=1$.
\end{problem}




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