Math 772

Spring 2002

Homework 3

Due Friday, March 8, 2002. Don’t forget to indicate your “favorite” problem.

1. Let be a Dedekind domain with two-generated ideal Show that the ideal (Note that the analog for principal ideals here is easier).

2. Let be a (quadratic) ring of integers and let be a nonzero ideal and let Show that can be generated by and some element (that is, ). (In particular, this problem shows that any ideal can be generated by two (or one) elements).

3. Let be a (quadratic) ring of integers and a nonzero ideal. We define the conjugate of to be . Show that the ideal is principal and generated by a rational integer (this ideal is called the norm of ).

4. Let be a square-free integer. We define the discriminant of to be . Let be the ring of integers of and a nonzero rational prime. Show that for some prime ideal if and only if divides the discriminant (such primes are said to be ramified and by the above there are only finitely many of them in a fixed ).