\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\\Summer 2006\\Homework 4} \date{} \maketitle \thispagestyle{empty} \pagestyle{empty} \centerline{\it Due Monday, July 24, 2006.} \centerline{} \noindent 1. Find the class group for each quadratic ring of integers below. \begin{itemize} \item[a)] (5 pt) $\mathbb{Z}[\sqrt{-14}]$. \item[b)] (5 pt) $\mathbb{Z}[\sqrt{-10}]$. \item[c)] (5 pt) $\mathbb{Z}[\frac{1+\sqrt{-23}}{2}]$. \item[d)] (5 pt) $\mathbb{Z}[\sqrt{-21}]$ \item[e)] (5 pt) $\mathbb{Z}[\frac{1+\sqrt{-163}}{2}]$. \end{itemize} \centerline{} \noindent 2. (5 pt) Find the smallest positive square-free integer, $d$, such that the ring of integers of the field $\mathbb{Q}[\sqrt{d}]$ is not a UFD. \centerline{} \noindent 3. (5 pt) Explicitly show for quadratic fields that a prime is ramified if and only if it divides the discriminant. \centerline{} \noindent 4. Consider the field $\mathbb{Q}(\alpha)$ where $\alpha$ is a root of $x^5-x^3+1$. You may assume that the ring of integers of $\mathbb{Q}(\alpha)$ is $\mathbb{Z}[\alpha]$. \begin{itemize} \item[a)] (5 pt) Find the number of real and complex embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$. \item[b)] (5 pt) Find the discriminant of the field $\mathbb{Q}(\alpha)$. \item[c)] (5 pt) Find the class group of the ring $\mathbb{Z}[\alpha]$. \item[d)] (5 pt) Determine how the ramified primes factor in $\mathbb{Z}[\alpha]$. \item[e)] (5 pt) Show that there is an element of norm 27 and of norm 9, but no element of norm 3. \end{itemize} \end{document}