\documentclass[12pt]{amsart}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=7in
\begin{document}
\author{}
\title{Math 421-621\\Fall 2013\\Homework 1}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}

\centerline{\it Due Wednesday, January 16, 2013.}


\centerline{}



\noindent 1. (5 pt) Classify all abelian groups of order $p^2q^3$ where $p,q$ are distinct positive primes.


\centerline{}

\noindent 2. (5 pt) Show that there is no simple group of order 50.

\centerline{}

\noindent 3. (5 pt) Let $G$ be a group and $G^{\prime}$ be its commutator subgroup. What can you say about the commutator subgroup of $G/G^{\prime}$?

\centerline{}

\noindent 4. (5 pt) Let $R$ be commutative with identity and $I\subseteq R$ an ideal such that 

\[
I:=\bigcap_{i\in \Gamma}\mathfrak{P}_i
\]

\noindent where each $\mathfrak{P}_i$ is a prime ideal. Show that $I$ is a radical ideal.

\centerline{}

\noindent 5. (5 pt) Let $d<0$ be a square free integer. Find the group of units of the domain $\mathbb{Z}[\sqrt{d}]$.

\centerline{}

\noindent 6. (5 pt) Show that the group of units of $\mathbb{Z}[\sqrt{2}]$ is infinite.

\centerline{}

\noindent 7. (5 pt) Let $R$ be a commutative ring with identity. We define the Jacobson radical of $R$, $J(R)$, to be the intersection of all of the maximal ideals of $R$. Show that $x\in J(R)$ if and only if $1+rx\in U(R)$ for all $r\in R$.


\centerline{}


\noindent 8. Give the definition for each of the following.
\begin{itemize}
\item[a)] (3 pt) What is an integral domain?
\item[b)] (3 pt) If $H\subseteq G$ is a group, what is the normalizer of $H$ is $G$?
\item[c)] (3 pt) What is a prime ideal?
\item[d)] (3 pt) What is a maximal ideal?
\item[e)] (3 pt) What is a multiplicatively closed subset of a ring?
\item[f)] (3 pt) What does it mean for a multiplicatively closed set to be saturated?
\end{itemize}



\end{document}