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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{}
\date{}


\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
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\newcommand{\Zi}{\Z[i]}

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\begin{document}
\maketitle
\vspace{-35pt}
\begin{center}
{\huge \bf Homework 2}\\\vspace{15pt}
{\Large Due Tuesday, October 19 in class.}
\end{center}

\vspace{30pt}
In Homework 1 we defined the number system $\Z[i]$, consisting of formal expressions of the form $a+bi$ where $a\in \Z$ and $b\in \Z$, and defined addition and multiplication.\\

Let $x,y\in \Z[i]$. Just as we did in Script 1, we say that $y$ divides $x$ (and that $y$ is a \emph{divisor} of $x$, and write $y|x$) if there is some $z\in \Z[i]$ such that $x=y\cdot z$.\vspace{10pt}
\begin{question}
Prove that any nonzero element of $\Z[i]$ has finitely many divisors in $\Z[i]$.
\end{question}\vspace{4pt}

\begin{question}

Let $x,y\in \Z[i]$. If $x|y$ and $y|x$, we say that $y$ is an \emph{associate} of $x$.\linebreak Prove that if $y$ is an associate of $x$, then $y$ is either equal to $x$, $-x$, $ix$, or $-ix$.
\end{question}\vspace{4pt}

\begin{question}[The Division Algorithm for $\Zi$]
\label{divisionalgorithm}
If $x,y\in \Z[i]$ and $y\neq 0$, then there exist $q\in \Z[i]$ and $r\in \Z[i]$ so that
\[x=yq+r\qquad\qquad\text{ and }\qquad\qquad N(r)<N(y).\]
\end{question}\vspace{0.5pt}

\begin{question}
Note that we did not require $q$ and $r$ to be unique in Question~\ref{divisionalgorithm}.\linebreak Give an example of elements $x\in \Z[i]$ and $y\in \Z[i]$ for which there is more than one possible choice of $q$ and $r$. Can you find an example where there are more than \emph{two} possible choices of $q$ and $r$?
\end{question}


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