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\prof{Peter Cholak and Juan Migliore}
\class{Math 222}
\quizdate{Friday, January 26}
\pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  It is suggested that you
first write out a draft of your proposed questions and then carefully
rewrite that draft to get your final version.  You do \emph{not} have
to write the answers on this sheet of paper, but please staple all
sheets together before turning your quiz in.\\

\begin{problem}
  Show by induction that $n^4 -4n^2$ is divisible by $3$ for all $n
  \geq 1$.
\end{problem}

\vskip 2in

\begin{problem}
In the ``real'' world, we are used to the fact that a power 
of a  non-zero number is again non-zero.  We will later see that this
fact is not true in all worlds.  Show that the complex numbers do
behave normally in this regard.  Precisely, let $z$ be an
arbitrary  complex number and suppose that we have $z^k = 0$ for some
positive integer $k$.  Show that $z$  must be zero.  (You may use the
fact that you know it to be true for real numbers.)
\end{problem}

\vskip 2in

\begin{problem}
  Find {\em all} the cube roots of $-8i$.  Write them in the the form
  $a+bi$.  Be sure to show your work.
\end{problem}

\end{document}
             
\end



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