Math 772

Spring 2002

Homework 2

Due Monday February 4, 2002.

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. We define a quadratic field as a field of the form where is a square-free integer (that is is not divisible by the square of any prime).

a) Show that every element of is the root of a polynomial of degree at most 2 with coefficients in

b) If is the set of elements of that are roots of a monic (that is leading coefficient 1) polynomial in then show is a ring between and that is not a field (this ring is called a ring of algebraic integers).

c) Show that

2. Show that the equation has no solutions in integers if 5 does not divide

3. Let be a quadratic field (as in problem #1). Find all square-free values of and integers such that contains a primitive root of 1 (hint: looking at primes and powers of 2 might be helpful, and recall that roots of unity must sit on the unit circle in the complex plane).