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\title{\vspace{-50pt}Elementary Number Theory\\
\normalsize Math 175, Section 30, Autumn 2010\\
\large Shmuel Weinberger ({\tt shmuel@math.uchicago.edu})\\
Tom Church ({\tt tchurch@math.uchicago.edu})\\
\url{www.math.uchicago.edu/~tchurch/teaching/175/}}
\author{}
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\begin{document}
\maketitle
\vspace{-35pt}
\begin{center}
{\huge \bf Homework 4}\\\vspace{15pt}
{\Large Due Tuesday, November 2 in class.\footnote{Election day!}}
\end{center}

\begin{question}Let $P(x)$ be a polynomial with integer coefficients of the form
\[P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0,\]
with $a_i\in \Z$ for each $i$.
\begin{enumerate}[a)]
\item Prove that if $r\in \Q$ is a root of $P(x)$\,---\,meaning that $P(r)=0$\,---\,then:
\begin{enumerate}[i)]
\item $r$ is an integer, and
\item $r|a_0$.
\end{enumerate}
\item If the degree $n$ is odd, the constant term $a_0$ is odd, and $P(x)$ has an odd number of odd coefficients,
\[\text{i.e. the size of the set\ \ }\big\{1\leq i\leq n-1\ \big|\ a_i\text{ is odd}\big\}\text{\ \ is odd}\qquad\qquad\]
then $P(x)$ has at least one irrational real root.

(Some examples of polynomials satisfying this condition are $x^3-2x^2+9x+1$ and $x^5-11x^4+7$.)
\end{enumerate}
\end{question}
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