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).

1. 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).

1. 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 ).

1. 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 ).