Math 129: Algebraic Number Theory Homework Assignment 2

William Stein

Due: Thursday, February 26, 2004

The Problems:

1. (a) Let k be a field. Prove that k[x] is a Dedekind domain.

(b) (Problem 1.12 from Swinnerton-Dyer) Let x be an indeterminate. Show that the ring Z[x] is Noetherian and integrally closed in its field of fractions, but is not a Dedekind domain.

2. Use MAGMA to write each of the following (fractional) ideals as a product of explicitly given prime ideals:

(a) The ideal (2004) in Q(\sqrt{−1}).

(b) The ideals I = (7) and J = (3) in the ring of integers of Q(ζ_{7}), where ζ_{7} is a root of the irreducible polynomial x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1. (The field Q(ζ_{7}) is called the 7th cyclotomic field.)

(c) The principal fractional ideal (3/8) in Q(\sqrt{5}).

3. Suppose R is an order in the ring O_{K} of integers of a number field. (Recall that an order is a subring of finite index in O_{K}.) For each of the following questions, either explain why the answer is yes for any possible order R in any O_{K}, or find one specific counterexample:

(a) Is R necessarily Noetherian?

(b) Is R necessarily integrally closed in its field of fractions?

(c) Is every nonzero prime ideal of R necessarily maximal?

(d) Is it always possible to write every ideal of R uniquely as a product of prime ideals of R?

4. Let O_{K} be the ring of integers of a number field K. Prove that the group of fractional ideals of O_{K}, under multiplication is (non-canonically) isomorphic to the group of positive rational numbers under multiplication.