\documentclass[12pt]{amsart}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\voffset=-1in
\hoffset=-1in
\textheight=10in
\textwidth=6in
\begin{document}
\author{}
\title{Math 420-620\\Fall 2012\\Homework 14}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}

\centerline{\it Due Friday December 7, 2012.}

\centerline{}


\noindent 1. Let $d\neq 1$ be a square-free integer and consider the set of elements $\mathbb{Q}[\sqrt{d}]=\{a+b\sqrt{d}\vert a,b\in\mathbb{Q}\}$. 
\begin{itemize}
\item[a)] (5 pt) Show that $\mathbb{Q}[\sqrt{d}]$ is a field. 
\item[b)] (5 pt) Show that $\mathbb{Q}[\sqrt{d}]=\mathbb{Z}[\sqrt{d}]_S$, where $S$ is the multiplicative set of nonzero integers.
\end{itemize}

\centerline{}

\noindent 2. (5 pt) Consider the function $\phi:\mathbb{Q}[\sqrt{d}]\longrightarrow\mathbb{Q}[\sqrt{d}]$ given by $\phi(a+b\sqrt{d})=a-b\sqrt{d}$. Show that $\phi$ is an automorphism of fields.


\centerline{}

\noindent 3. Let $R:=\mathbb{Z}[\sqrt{d}]=\{m+n\sqrt{d}\vert m,n\in\mathbb{Z}\}$ and consider the function $N:\mathbb{Z}[\sqrt{d}]\longrightarrow\mathbb{Z}$ given by $N(m+n\sqrt{d})=m^2-dn^2$. Show that the function $N$ enjoys the following properties.
\begin{itemize}
\item[a)] (5 pt) $N(\alpha\beta)=N(\alpha)N(\beta),$ for all $\alpha,\beta\in R$.
\item[b)] (5 pt) $N(\alpha)=0$ if and only if $\alpha=0$.
\item[c)] (5 pt) $N(\alpha)=\pm 1$ if and only if $\alpha\in U(R)$.
\end{itemize}



\centerline{}


\noindent 4. (5 pt) Show that the Gaussian integers $\mathbb{Z}[i]$ is Euclidean. What can you say about factorization in $\mathbb{Z}[i]$?

\centerline{}


\noindent 5. Consider the integral domain $\mathbb{Z}[\sqrt{-5}]$, and $\mathbb{Z}[\sqrt{-14}]$.
\begin{itemize}
\item[a)] (5 pt) Show that the elements $2,3,1+\sqrt{-5},$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$
\item[b)] (5 pt) Show that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD.
\item[c)] (5 pt) Find an element with two irreducible factorizations of different lengths in $\mathbb{Z}[\sqrt{-14}]$.
\end{itemize}






\end{document}