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\noindent{\Large{\textsc{Complex Variables\hfill Assignment 2.2  \\ Spring 2020 \hfill Due February 5}}}

\large

\vspace{1cm}


\begin{exercise}
Let $n \in \N$.  

\begin{itemize}

\item[\bf a.] Explain why $\displaystyle X^n - 1 = \prod_{\zeta \in \mu_n} (X  - \zeta)$.


\item[\bf b.]  Use part {\bf a} to prove that $\displaystyle \prod_{\zeta \in \mu_n} \zeta = (-1)^{n+1}$ and $\displaystyle \sum_{\zeta \in \mu_n} \zeta = 0$ (for $n \ge 2$).


\end{itemize}

\end{exercise}

\begin{exercise}  
Prove that for $n \in \N$ and $k \in \Z$ one has
\[
\sum_{\zeta \in \mu_n} \zeta^k = \begin{cases} n, & \text{if } k \equiv 0 \!\!\!\! \pmod{n}, \\ 0, & \text{otherwise.}  \end{cases}
\]
[{\em Suggestion:}  Use the fact that $\mu_n = \{ 1, \omega, \omega^2, \ldots, \omega^{n-1}\}$, where $\omega = e^{2 \pi i / n}$.]
\end{exercise}

\begin{exercise}
For $n \in \N$, let $f(X) = X^n - 1$ and $g(X) = X^{n-1} + X^{n-2} + \cdots + X + 1$ so that $f(X) = (X-1) g(X)$.  Suppose $\zeta \in \mu_n$, $\zeta \ne 1$.

\begin{itemize}

\item[\bf a.] Prove that $f'(\zeta) = (\zeta - 1)g'(\zeta)$.  Conclude that $\displaystyle g'(\zeta) = \frac{n \zeta^{n-1}}{\zeta - 1}$.

\item[\bf b.]  Use part {\bf a} to show that $\displaystyle n\zeta^{n-1} + (n-1) \zeta^{n-2} + \cdots + 2 \zeta + 1 = \frac{n}{\zeta - 1}$.

\end{itemize}

\end{exercise}

\begin{exercise}
Compute $\mu_8$.
\end{exercise}

\begin{exercise}
Find the 3rd roots of $-8$.
\end{exercise}

\begin{exercise}
Find the 4th roots of $a=3 - 4i$.  [{\em Warning:}  Double check the sign of the imaginary part of your answer. ]
\end{exercise}

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