\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage[all]{xy}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=6in
\begin{document}
\author{}
\title{Math 772\\Spring 2011\\Homework 2}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Monday October 2, 2011.}




\centerline{}



\noindent 1. Find all primes $p$ such that:
\begin{itemize} 
\item[a)] (5 pt) $-7$ is a square $\text{mod}(p)$.
\item[b)] (5 pt) $\frac{2}{5}$ is a square $\text{mod}(p)$.
\end{itemize}

\centerline{}

\noindent 2. (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 3. (5 pt) Consider the family of quadratic rings of integers $R:=\mathbb{Z}[\omega]$ where $\omega$ is given by
\[
\omega=
\begin{cases}
\sqrt{d},\text{ if }d\equiv 2,3\text{ mod}(4);\\
\frac{1+\sqrt{d}}{2},\text{ if }d\equiv 1\text{ mod}(4).
\end{cases}
\]

 Show that if $R$ is an imaginary quadratic ring of integers ($d<0$), 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.

\centerline{}
\noindent 4. (5 pt) Let $R$ be an integral domain with quotient field $K$. We define the {\it integral closure} of $R$ to be 

\[
\overline{R}=\{\alpha\in K\vert p(\alpha)=0\text{ for some monic }p(x)\in R[x]\}.
\]

\noindent We say that $R$ is integrally closed if $R=\overline{R}$ (that is, $R$ already contains all of its integral elements from $K$). Prove that any UFD is integrally closed.

\centerline{}

\noindent 5. (5 pt) Let $R$ be an integral domain with quotient field $K$ and $p\in R$ a nonzero prime element. Show that $p$ is also a prime element of $R[x]$.

\centerline{}

\noindent 6. (5 pt) Let $F$ be a field extension of degree $n$ over $\mathbb{Q}$. Suppose that $\omega\in F$ is a root of a monic polynomial in $\mathbb{Z}[x]$. Show that $\omega$ is a root of a monic polynomial in $\mathbb{Z}[x]$ of degree no more than $n$ and, in particular, show that the minimal polynomial of $\omega$ (over $\mathbb{Q}$) may be taken to be monic and in $\mathbb{Z}[x]$.

\centerline{}


\noindent 7. (5 pt) Let $d$ be a square-free integer. Show that the ring of integers of the quadratic field $\mathbb{Q}(\sqrt{d})$ is given by

\[
R=\begin{cases}

\mathbb{Z}[\sqrt{d}] & \text{if $d\equiv 2,3\ \text{mod}(4)$,}\\
\mathbb{Z}[\frac{1+\sqrt{d}}{2}] & \text{if $d\equiv 1\ \text{mod}(4)$.}
\end{cases}
\]




\end{document}