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

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\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) $\sqrt[6]{4}$\quad (over the complex numbers).\\
b) The {\em argument} of a complex number $z$.\\
c) The {\em order} of an $n$th root of unity for a positive integer $n$.\\
d) The {\em additive inverse} of a number $a$ in $\mathbb{Z}_n$ ($n$ a
postive integer).
\end{problem}
\\
\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
required:\\
a) $\operatorname{arg}(1 + i) = \pi/4 + 2{\pi}n$ for any $n \in \mathbb{Z}$.\\
b) For any $z \in \mathbb{C}$, $|z|^3 = 3|z|$.\\
c) $3\cdot17\equiv 3 \quad (\operatorname{mod} 6)$.\\
d) The coefficient of $a^7b^4$ in $(a + b)^{11}$ is 330.
\end{problem}
\\
\begin{problem}(15 points) Find all solutions to $x^5-32=0$. Are these
  solutions constructible? Why or why not? (You do not need to simplify
your answer to the
   form $a + bi$.)
\end{problem}
\\
\begin{problem}(15 points) Find the multiplicative inverse of 14 in
$\mathbb{Z}_{59}$.
\end{problem}
\\
\begin{problem}(15 points) Prove that $n^{13} - n$ is divisible by $78$ for
any positive
integer $n$.
\end{problem}
\\
\begin{problem}(15 points) Exactly one of the following three problems will
appear.

a) Let $\operatorname{lcm}(a,b)$ denote the {\em least common multiple} of
a pair of integers
$a,b$. Let $\zeta$ be the first $n$th root of unity for some positive
integer $n$ (i.e.\
$\zeta \neq 1$).  Show that $\displaystyle o(\zeta^k) =
\frac{\operatorname{lcm}(k,n)}{k}$ for
$0 < k < n$.\\

b) 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 $\mathbb{Z}_{n}$.\\

c) Prove that if $n$ is an odd positive integer and $\zeta$ is any
primitive $n$th root of
unity, then so is $\zeta^2$.  Is this also true for $n$ even?  Why or why not?
\end{problem}
\newpage
\begin{problem}(15 points) Exactly one of the following three problems will
appear.

a) Prove that the least common multiple of any two positive integers $k$
and $m$ is
$\displaystyle\frac{km}{(k,m)}$.\\

b) Let $p, p_1, p_2, p_3 \in \mathbb{Z}$ be prime numbers.  Use Euclid's
Lemma (Lemma 4.4) to
prove that if $p$ divides $n = p_1\cdot p_2\cdot p_3$, then $p$ equals one
of $p_1$, $p_2$, or
$p_3$.\\

c) 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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