Harvard Math 129: Algebraic Number Theory Homework Assignment 2

William Stein

Due: Thursday, February 24, 2005

The problems have equal point value, and multi-part problems are of the same value. There are 7 problems.

1. Prove that the ring \overline{ℤ} is not noetherian, but it is integrally closed in its field of fraction, and every nonzero prime ideal is maximal. Thus \overline{ℤ} is not a Dedekind domain.

2. Let K be a field.

(a) Prove that the polynomial ring K[x] is a Dedekind domain.

(b) Is ℤ[x] a Dedekind domain?

3. Let O_{K} be the ring of integers of a number field. Let F_{K} denote the abelian group of fractional ideals of O_{K}.

(a) Prove that F_{K} is torsion free.

(b) Prove that F_{K} is not finitely generated.

(c) Prove that F_{K} is countable.

(d) Conclude that if K and L are number fields, then there exists an isomorphism of groups F_{K} ≈ F_{L}.

4. From basic definitions, find the rings of integers of the fields ℚ(\sqrt{11}) and ℚ(\sqrt{13}).

5. Factor the ideal (10) as a product of primes in the ring of integers of ℚ(\sqrt{11}). You’re allowed to use a computer, as long as you show the commands you use.

6. Let O_{K} be the ring of integers of a number field K, and let p ∈ ℤ be a prime number. What is the cardinality of O_{K}/(p) in terms of p and [K : ℚ], where (p) is the ideal of O_{K} generated by p? (Prove that your formula is correct.)

7. Give an example of each of the following, with proof:

(a) A non-principal ideal in a ring.

(b) A module that is not finitely generated.

(c) The ring of integers of a number field of degree 3.

(d) An order in the ring of integers of a number field of degree 5.

(e) The matrix on K of left multiplication by an element of K, where K is a degree 3 number field.

(f) An integral domain that is not integrally closed in its field of fractions.

(g) A Dedekind domain with finite cardinality.

(h) A fractional ideal of the ring of integers of a number field that is not an integral ideal.