Math 772

Spring 2002

Homework 1

(Turn in by Wednesday January 23, 2002).

First a note on how the homeworks/exams will go. On each homework, there will be n problems (all of which should be turned in). On the first page of the assignment you should select one problem that will be graded on a 0-10 scale (your favorite problem, if you will). The remainder will be graded on a 0-3 scale. (In all cases “0” means that you did not do it, so you should at least attempt all of the problems). For exams all problems will be on a 0-10 scale.

1. Prove that every element of a finite field can be written as the sum of two squares.

2. Show that if is a positive prime integer, then (This result is known as Wilson’s Theorem).

3. Let and be a positive prime integer. Prove the “Freshman’s Dream,” that is,

4. Determine all positive prime integers such that:

a) –2 is a square mod

b) 3 is a square mod

c) –6 is a square mod