\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 0}
\date{}
\maketitle
\thispagestyle{empty}
\pagestyle{empty}



\centerline{\it Due Wednesday June 14, 2006.}

\centerline{}

\noindent 1. (5 pt) Let $\mathbb{F}$ be a field and let $\mathfrak{F}$ be a (multiplicative) subgroup of $\mathbb{F}\setminus\{0\}$. Show that if $\mathfrak{F}$ is finite, then $\mathfrak{F}$ is cyclic.

\centerline{}

\noindent 2. (5 pt) Let $\mathbb{F}$ be a finite field. Show that every element in $\mathbb{F}$ can be written as the sum of two squares (that is, if $a\in\mathbb{F}$ then $a=x^2+y^2$ for some $x,y\in\mathbb{F}$). Is this result true if the word ``finite" is removed?

\centerline{}

\noindent 3. (5 pt) Let $p$ be an odd prime and $\mathbb{F}$ be the finite field of $p^n$ elements. Show that $-1$ is a square in $\mathbb{F}$ (that is, $-1=x^2$ for some $x\in\mathbb{F}$) if and only if $p^n\equiv 1\text{ mod}(4)$. Use this to find all odd primes for which $-1$ is a square mod($p$). What happens if $p=2$?



\end{document}