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\begin{center} {\Large\bf Honors Algebra I\\ Assignment 12} \\ Not
to be Submitted {\em Happy Thanksgiving}
\end{center}
\begin{quote}
I was 21 years when I wrote this song
\\ I'm 22 now, but I won't be for long
\\ Time hurries on
\\ And the leaves that are green turn to brown
\\ -- Paul Simon, {\em Leaves That are Green}
\end{quote}
These problems concern the Gaussian Integers $Z[i]$ with
$d(\ah)=\ah\ol{\ah}=a^2+b^2$ where $\ah=a+bi$ and $\ol{\ah}$
represents the complex conjugate $a-bi$. By prime we mean in
the integers $Z$ but by Gaussian prime we mean in $Z[i]$. For
example, $5$ is a prime but not a Gaussian prime as $5=(2+i)(2-i)$.
When we say ``factors into $s$ primes" below, we count with repetition,
so that $12=2\cdot 2\cdot 3$ factors into $3$ primes. We write $x\sim y$
if $x=yu$ with $u$ a unit.
\ben
\item Show that $\gam$ is a unit if and only if $\ol{\gam}$ is
a unit.
\item Show that $\ah$ is a Gaussian prime if and only if $\ol{\ah}$ is
a Gaussian prime.
\item There is precisely one Gaussian prime $\ah$ such that $\ah\sim \ol{\ah}$,
other that those which are $\sim p$ for some integer $p$. Which $\ah$ is it,
and why is it the only one?
\item Show that $\ah$ factors into $t$ Gaussian primes if and only if $\ol{\ah}$
factors into $t$ Gaussian primes.
\item Show that if $p$ is a prime then either it is a Gaussian Prime or
$p=\ah\ol{\ah}$ where $\ah,\ol{\ah}$ are Gaussian Primes.
\item Show that if $d(\ah)$ is a prime then $\ah$ is a Gaussian prime.
\item Show that if $n$ factors into $s$ primes then it factors into
{\em at least} $s$ Gaussian primes.
\item Show that if $d(\ah)$ factors into three or more primes then
$\ah$ cannot be a Gaussian prime. (Idea: look at the factorization of
$d(\ah)=\ah\ol{\ah}$.)
\item Suppose $d(\ah)=pq$ where $p,q$ are distinct primes. Show that
$\ah$ cannot be a Gaussian prime.
\item Suppose $p$ is a prime and there is no expression $p=x^2+y^2$ with
$x,y\in Z$. Show that $p$ is a Gaussian prime.
\item Suppose $p$ is a prime and there is an expression $p=x^2+y^2$ with
$x,y\in Z$. Show that $p$ is a not a Gaussian prime.
\item Show that if $d(\ah)=p^2$ and $\ah$ is a Gaussian prime then it
must be that $\ah\sim p$ and there is no expression $p=x^2+y^2$ with
$x,y\in Z$.
\item Give the prime factorizations $5,13,17$ in $Z[i]$.
Set $n=1105=5\cdot 13 \cdot 17$.
Give the prime factorization of $n$ in $Z[i]$.
Use this to find four ``distinct" ways to write $n=\beta\ol{\beta}$.
(That is, switching the order or multiplying by units gives the ``same" way.)
Use this to find four expressions of $n$ as the sum of two squares.
\item Generalizing: Suppose $n$ is the product of $s$ distinct odd primes
$p_1,\ldots,p_s$, each of which is not a Gaussian Prime. Show that
$n$ can be written as the sum of two squares in $2^{s-1}$ different ways.
\een
\begin{quote}
Homeward bound
\\ I wish I was
\\ Homeward bound
\\ Home, where my thoughtâ€™s escaping
\\ Home, where my musicâ€™s playing
\\ Home ,where my love lies waiting
\\ Silently for me
\\ -- Paul Simon, {\em Homeward Bound}
\end{quote}
\end{document}