\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\\Fall 2011\\Homework 3}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Monday, October 24, 2011.}

\centerline{}

\noindent 1. Consider the rings of integers $\mathbb{Z}[\sqrt{2}]$ and $\mathbb{Z}[\sqrt{-2}]$.
\begin{itemize}
\item[a)] (5 pt) Pick one of these domains and show that it is Euclidean (they both are and you may use this knowledge in the next two parts).
\item[b)] (5 pt) Find all positive primes that can be represented in the form $x^2+2y^2$.
\item[c)] (5 pt) Find all positive primes that can be represented in the form $x^2-2y^2$.
\end{itemize}

\centerline{}

\noindent 2. Consider the ring of algebraic integers $R$ with quotient field $K$. One can define the norm ($N$) from $K$ to $\mathbb{Q}$ via

\[
N(x)=\prod_{\sigma}\sigma(x)
\]

\noindent where the product ranges over all the distinct embeddings of $K$ into $\mathbb{C}$ (and the norm on $R$ is just the restriction of this map to $R$).

Perform the following tasks.
\begin{itemize}
\item[a)] (5 pt) Show $N(xy)=N(x)N(y)$ (this should be easy!).
\item[b)] (5 pt) Show $N(x)=0$ if and only if $x=0$.
\item[c)] (5 pt) Show that the image of the norm is contained in $\mathbb{Q}$ and if $x\in R$ then $N(x)\in\mathbb{Z}$.
\item[d)] (5 pt) Show that if $x\in R$ then $x$ is a unit of $R$ if and only if $N(x)=\pm 1$.
\item[e)] (5 pt) For the case $R=\mathbb{Z}[\sqrt[3]{2}]$ explicitly find the norm function.
\item[f)] (5 pt) Use the previous part to show that $\mathbb{Z}[2\sqrt[3]{2}]$ is not an HFD.
\end{itemize}

\centerline{}

\noindent 3. (5 pt) Determine a quadratic ring of integers such that the inert primes are precisely the primes $p$ such that $p\equiv 5,11\ \text{mod}(12)$ or convince me that there is no such animal.

\centerline{}



\end{document}