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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{}


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\begin{document}
\maketitle
\vspace{-40pt}
\begin{center}
{\huge \bf Homework 6}\\\vspace{15pt}
{\Large Due Thursday, November 18 in class.}
\end{center}

\begin{question}
The primes less than 100 are:  \[2, 3,      5,      7,     11,     13,     17,     19,     23,     29, 
     31,     37,     41,     43,     47,     53,   59,     61,     67,     71, 
     73,     79,     83,     89,\text{ and }    97.\]
%    101    103    107    109    113     127    131    137    139    149    151    157    163    167    173     179    181    191    193    197    199

Those that are congruent to $1\pmod{4}$ are \textbf{5},     \textbf{13},    \textbf{ 17},      \textbf{ 29}, 
     \textbf{ 37},     \textbf{41},       \textbf{53},     
     \mbox{\textbf{61}, \textbf{73},       \textbf{89}, and     \textbf{97}.}
Write each of these primes as a sum of two squares. For example,  $13 = 9+4=3^2+2^2$. (The squares less than 100 are 1, 2, 4, 9, 16, 25, 36, 49, 64, and 81.)
\end{question}\vskip20pt

We proved in Theorem 3.31  that any prime $p$ such that $p\equiv 1\pmod{4}$ can be written as $p=a^2+b^2$ for some $a$ and $b$. In the rest of this homework you will use Gaussian integers to give a quicker and simpler proof of this theorem. You will also prove that $p$ can be \emph{uniquely} expressed as the sum of two squares, which we did not prove in class.

\begin{question}
Prove that if $p$ is a prime with $p\equiv 1 \pmod{4}$, the ring $\Z[i]/(p)$ is not a field. (Hint: find more than two roots in $\Z[i]/(p)$ of the polynomial $P(x)=x^2+1$.)
\end{question}

\begin{question}
Prove that if $p$ is a prime in $\Z$ and there is no element $x\in \Z[i]$ with norm $N(x)=p$, then $p$ is irreducible in $\Z[i]$.
\end{question}

\begin{question}
Using Questions 2 and 3, prove that if $p$ is a prime with $p\equiv 1 \pmod{4}$, then $p$ can be written as $p=a^2+b^2$ for some $a,b\in \Z$.
\end{question}\vskip20pt

\begin{question}
Prove that $p$ can be \emph{uniquely} written as $p=a^2+b^2$: if we can also write $p=c^2+d^2$, then either $a=\pm c$ and $b=\pm d$, or vice versa.
\end{question}

\begin{question}
Let $p$ and $q$ be primes with $p\equiv 1\pmod{4}$ and $q\equiv 1\pmod{4}$. How many distinct ways are there of writing $pq=a^2+b^2$? (Assume  $a>b>0$ to avoid double-counting.)
\end{question}

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