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\begin{center}
{\large MATH497C Assignment 10} \\
{\bf Due: November 30, 2012}
\end{center}
\begin{problem}*
Let $p$ be an {\it irreducible} integral polynomial (ie, $p$ cannot be written as $p = p_1p_2$, where $p_1$ and $p_2$ are polynomial of degree greater than 1 with coefficients in $\Q$). Show that $p$ must have distinct roots (ie, each root appears with multiplicity 1).
\end{problem}
\begin{problem}
Show that if $p$ is an integral polynomial and the roots of $p$ all have modulus 1, then they are all roots of unity. Find an integer polynomial which has roots of modulus 1 which are not roots of unity.
\end{problem}
\begin{problem}
This problem deals with {\it topological mixing}. A system $f : X \to X$ is topogically mixing if for every pair of nonempty open sets $U$ and $V$, there exists $N \in \N$ such that if $n \ge N$, $f^n(U) \cap V \not= \emptyset$. Compare this with the corresponding definition for topological transitivity. Prove that the automorphism of $\T^2$ induced by the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ is topologically mixing using the following steps:
\begin{enumerate}[(a)]
\item A set $A\subset X$ is called $\ve$-dense if for every $x \in X$, there exists a $y \in A$ such that $d(x,y) < \ve$. Show that a set $A$ is $\ve$-dense if and only if $B_\ve(A) = X$. Also, show that a set $A$ is dense if and only if it is $\ve$-dense for every $\ve$.
\item Let $L$ be any line in $\R^2$ not intersecting $\Z^2$, and $J \subset L$ be an interval contained in $L$. There is a natural notion of the length of $J$, defining $\abs{J}$ to be the distance between the endpoints (in $\R^2$). Show that for any $\ve > 0$, there exists a $C > 0$ such that if $J \subset L$ is a subinterval with $\abs{J} \ge C$, then $p(x+J)$ is $\ve$-dense for every $x \in \T^2$
\item Show that any open set $U$ contains a set of the form $J^s + J^u = \set{x^s + x^u : x^s \in J^s, x^u \in J^u}$, where $J^* \subset E^*$ for $* = s,u$, and $E^s,E^u$ are the eigenspaces as described in the lectures and notes for the map $A$
\item Use this to show that for any fixed open set $U$ and $\ve > 0$, there exists an $N \ge 0$ such that if $n \ge N$, $A^n(U)$ is $\ve$-dense
\item Show that $A$ is topologically mixing
\end{enumerate}
\end{problem}
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