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\title{Math 772\\Summer 2006\\Homework 1}
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\centerline{\it Due Monday June 26, 2006.}

\centerline{}

\noindent 1. (5 pt) Let $p$ be a prime integer. Characterize (and find the number of) units in the ring $\mathbb{Z}/p^n\mathbb{Z}$. Use this to find a formula for the number of units in $\mathbb{Z}/n\mathbb{Z}$ where $n>2$.

\centerline{}

\noindent 2. Find all primes $p$ such that:
\begin{itemize} 
\item[a)] (5 pt) $-7$ is a square $\text{mod}(p)$.
\item[b)] (5 pt) $\frac{3}{5}$ is a square $\text{mod}(p)$.
\end{itemize}

\centerline{}

\noindent 3. (5 pt) Show that if $p$ is an odd prime then the units of $\mathbb{Z}/p^n\mathbb{Z}$ form a cyclic group. What happens in the case that $p=2$ (what is the group structure of $U(\mathbb{Z}/2^n\mathbb{Z})$)? 

\centerline{}

\noindent 4. Let $d$ be a square-free integer (that is, $d$ is divisible by no square except 1) and consider the ring $R:=\mathbb{Z}[\omega]$ where $\omega$ is given by

\[
\omega=
\begin{cases}
\sqrt{d}\text{ if } d\equiv 2,3\text{mod}(4)\\
\frac{1+\sqrt{d}}{2}\text{ if } d\equiv 1\text{mod}(4)
\end{cases}
\]

\noindent These rings are called the {\it quadratic rings of integers}. If $d>0$ the quadratic ring of integers is called {\it real} and if $d<0$ then the quadratic ring of integers is called {\it imaginary}.

We define the {\it norm} ($N$) by $N(a+b\omega)=(a+b\omega)(a+b\overline{\omega})$ where 

\[
\overline{\omega}=
\begin{cases}
-\sqrt{d}\text{ if } d\equiv 2,3\text{mod}(4)\\
\frac{1-\sqrt{d}}{2}\text{ if } d\equiv 1\text{mod}(4)
\end{cases}
\]

\noindent Verify the following properties of the norm.

\begin{itemize}
\item[a)] (5 pt) $N(R)\subseteq\mathbb{Z}$.
\item[b)] (5 pt) $N(x)=0$ if and only if $x=0$.
\item[c)] (5 pt) $N(xy)=N(x)N(y)$.
\item[d)] (5 pt) $x\in U(R)$ if and only if $N(x)=\pm 1$.
\end{itemize}

\centerline{}

\noindent 5. (5 pt) Consider the family of quadratic rings of integers defined above. Show that if $R$ is an imaginary quadratic ring of integers, then $U(R)=\pm 1$ unless $d=-1$ or $d=-3$. What happens in these last two cases? By way of contrast, show that $U(\mathbb{Z}[\sqrt{2}])$ is infinite.


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