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\author{Math 25}

\title{Homework 9 --- due Nov 21st}

\begin{document}

\maketitle
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There are two sections: A and B.
Please return each part
to the proper CA.

In all these problems, if $R$ is a ring, $R^*=R \setminus \{0\}$.

\renewcommand{\thesection}{\Alph{section}}

\section{Problems graded by Benjamin}

\subsection{} Let $(E,d)$ be a metric space, and let $f:E \ra \CC$ be
a function. Write $f(x)=a(x)+i b(x)$ where $a,b$ are two functions
from $E$ to $\RR$. 
\begin{enumerate}
\item prove that $f$ is continuous if and only if $a$ and $b$ are
  continuous. 
\item prove that if $f,g$ are two continuous functions from $(E,d)$ to
  $\CC$ then $f \pm g$ and $f \cdot g$ are also continuous.
\end{enumerate} 

\subsection{} 
If $m \in \ZZ_{\geq 1}$ let $\phi(m) = \on{card} \{ 1 \leq x \leq m, \
\on{gcd}(x,m)=1 \}$. If $G$ is a group and if $g \in G$, the order of
$g$ is the smallest $m \geq 1$ such that $g^m=e$. Don't confuse this
with the order of $G$ which is $\on{card} G$.
\begin{enumerate}
\item prove that the order of $g$ is the order of the subgroup it
  generates in $G$ (assume that $G$ is finite).
\item prove that if $G$ is any finite group and if $g \in
  G$ then the order of $g$ divides the order of $G$.
\item prove that if $n \geq 1$, then $\sum_{d | n} \phi(d)=n$ (here
  $\sum_{d | n}$ means the sum over all divisors $d$ of $n$).
\item show that if $G$ is a cyclic group of order $n$ and $d$ divides
  $n$, then there are $\phi(d)$ elements in $g$ of order $d$.
\item in (5)-(7) let $K$ be a field and let $G$ be a finite subgroup of
  $(K^*,\times)$ of order $n$. Prove that there are at most $d$ elements $g \in G$
  such that $g^d=1$. 
\item prove that there exists an element $g \in G$ which is of order $n$.
\item prove that $G$ is cyclic.
\item let $G$ be the set of invertible elements in $\ZZ/24\ZZ$. Prove
  that $G$ is not a cyclic group.
\end{enumerate} 


\subsection{} Let $S^1$ be the set of complex numbers $z$ such that
$|z|=1$ and let $C_n$ be the set of complex numbers such that
$z^n=1$. 
\begin{enumerate}
\item prove that $(S^1,\times)$ and $(C_n,\times)$ are  compact
  subgroups of $(\CC^*,\times)$. 
\item are there any other ones?
\end{enumerate}

\subsection{} If $P(X)=X^d+a_{d-1}X^{d-1}+\cdots+a_0 \in \CC[X]$, let
  $R$ be the largest of the $|z_i|$ where $z_1,\cdots,z_d$ are the
  roots of $P$ in $\CC$. Prove the following inequalities:
\begin{enumerate}
\item if $r>0$ satisfies $r^d \geq \sum_{i=0}^{d-1} |a_i|r^i$,
  then $R \leq r$.
\item $R\leq \max(1,\sum_{i=1}^{d-1} |a_i|)$ (Montel)
\item $R \leq 1 + \max_{1 \leq k \leq d} |a_k|$  (Cauchy)
\item $R \leq |1-a_{d-1}|+|a_{d-1}-a_{d-2}|+\cdots+|a_1-a_0|+|a_0|$ (Montel)
\item if $a_i \in \RR$ for all $i$ and $1 \geq a_{d-1} \geq a_{d-2} \geq
  \cdots \geq a_0 \geq 0$ then $R \leq 1$ (Kakeya)
\end{enumerate}

\section{Problems graded by Inna}

\subsection{} Find a formula for $\sum_{k=0}^n \cos(kx)$ and for
$\sum_{k=0}^n \sin(kx)$. Hint: consider $1+e^{ix} + \cdots  +
e^{inx}$.

\subsection{} Prove that if $P \in \CC[X]$ is a non-constant
  polynomial, then the function $z \mapsto |P(z)|$ has no local
  maximum in $\CC$ (a local maximum is a point $z_0 \in \CC$ such that
  there exists $\delta >0$ with the property: if $|z-z_0|<\delta$,
  then $|P(z)| \leq |P(z_0)|$).

\subsection{} Let $P(X)=a_0+a_1 X +\cdots+a_d X^d \in \RR[X]$ be a
polynomial of degree $d$. We say that $P$ is 
split in $\RR$ if one can write $P(X)=a_d (X-\alpha_1) \cdots
(X-\alpha_d)$ where the $\alpha_i$'s are (not necessarily distinct)
elements of $\RR$. For example $X^2-2X+1$ is split but not $X^2+1$.
\begin{enumerate}
\item prove that if $P$ is split then $P'$ is split.
\item more generally, prove that if $\lambda \in \RR$, and if $P$ is
  split, then $P' + \lambda P$ is split (hint: consider the function
  $x \mapsto P(x)e^{\lambda x}$).
\item prove that if a polynomial 
$b_0+b_1 X +\cdots+b_d X^d$ and $P(X)$ are
both split then so is the polynomial 
$b_0 P(X) + b_1 P'(X) +\cdots + b_d P^{(d)}(X)$.
\end{enumerate}

\subsection{} If $z_1,\cdots,z_d$ are complex numbers, the convex hull
  of the $z_i$'s is the set of complex numbers of the form
  $z=\lambda_1 z_1 + \cdots + \lambda_d z_d$ where $\lambda_i \in
  \RR_{\geq 0}$ and $\lambda_1+\cdots+\lambda_d =1$. 
\begin{enumerate}
\item for $n=2,3,4,5$ describe the convex hull of the $n$ solutions of
  the equation $z^n=1$ (draw a picture!).
\item Let $P \in \CC[X]$ be a polynomial of degree $d$, whose roots
  are $z_1,\cdots,z_d$. Prove that
\[ \frac{P'(X)}{P(X)} = \sum_{i=1}^d \frac{1}{X-z_i} \]
\item use this to show that the roots of $P'(X)$ lie in the convex
  hull of the roots of $P(X)$.
\end{enumerate}
\end{document}

