\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\\Spring 2011\\Homework 1} \date{} \maketitle \thispagestyle{empty} \pagestyle{empty} \centerline{\it Due Wednesday September 14, 2011.} \centerline{} \noindent 1. In this problem, we will find all solutions to the equations $x^2+y^2=z^2$ with $x,y,z\in\mathbb{N}$ and look at some of the properties of the solutions. \begin{itemize} \item[a)] (5 pt) Show that if $(a,b,c),(u,v,w)\in\mathbb{Z}^3$ are solutions to $x^2+y^2=z^2$, then so are $(ta,tb,tc)$ and $(au-bv,av+bu,cw)$ (so the set of integral solutions forms a monoid with identity $(1,0,1)$). \item[b)] (5 pt) Find all rational solutions to the equations $u^2+v^2=1$ (hint: use the point $(-1,0)$ on the unit circle to parametrize the set of these solutions, or perhaps examine the Weierstrauss substitution from Calculus II). \item[c)] (5 pt) Show that the set of solutions to $x^2+y^2=z^2$ (with $x,y,z\in\mathbb{N}$) can be parametrized as follows \begin{align} & x=n^2-m^2\notag\\ & y=2nm\notag\\ & z=n^2+m^2\notag \end{align} \noindent with $x,y,z$ pairwise relatively prime. \item[d)] (5 pt) Show that exactly one of $n,m$ is odd and the other is even. \end{itemize} \centerline{} \noindent 2. ({\it Infinite Descent}) The objective of this problem is to show that the equation $x^4+y^4=z^4$ has no solutions for $x,y,z\in\mathbb{Z}\setminus\{0\}$. Note first that by the symmetry of this equation, we can assume that $x,y,z>0$. \begin{itemize} \item[a)] (5 pt) Consider first the equation $x^4+y^4=Z^2$. Use the results of the first problem to write $x^2,y^2,$ and $Z$ parametrically. Then find a Pythagorean triple involving $y$. \item[b)] (5 pt) Use the Pythagorean triple to find a square smaller than $Z^2$ that is the sum of two fourth powers. \item[c)] (5 pt) Derive a contradiction and explain why the equation $x^4+y^4=z^4$ has no nontrivial solution. \end{itemize} \centerline{} \noindent 3. (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 4. (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}