Harvard Math 129: Algebraic Number Theory Homework Assignment 3

William Stein

Due: Thursday, March 3, 2005

The problems have equal point value, and multi-part problems are of the same value. There are five problems. In any problem where you use a computer, include in your solution the exact commands you type and their output. You may use any software, including (but not limited to) MAGMA and PARI.

1. Let φ : R → S be a homomorphism of (commutative) rings.

(a) Prove that if I ⊂ S is an ideal, then φ^{−1}(I) is an ideal of R.

(b) Prove moreover that if I is prime, then φ^{−1}(I) is also prime.

2. Let O_{K} be the ring of integers of a number field. The Zariski topology on the set X = Spec(O_{K}) of all prime ideals of O_{K} has closed sets the sets of the form

V (I) = {𝔭 ∈ X : 𝔭 | I},

where I varies through all ideals of O_{K}, and 𝔭 | I means that I ⊂ 𝔭.

(a) Prove that the collection of closed sets of the form V (I) is a topology on X.

(b) Let Y be the subset of nonzero prime ideals of O_{K}, with the induced topology. Use unique factorization of ideals to prove that the closed subsets of Y are exactly the finite subsets of Y along with the set Y.

(c) Prove that the conclusion of (a) is still true if O_{K} is replaced by an order in O_{K}, i.e., a subring that has finite index in O_{K} as a ℤ-module.

3. Explicitly factor the ideals generated by each of 2, 3, and 5 in the ring of integers of ℚ(\sqrt[3]{2}). (Thus you’ll factor 3 separate ideals as products of prime ideals.) You may assume that the ring of integers of ℚ(\sqrt[3]{2}) is ℤ[\sqrt[3]{2}], but do not simply use a computer command to do the factorizations.

4. Let K = ℚ(ζ_{13}),where ζ_{13} is a primitive 13th root of unity. Note that K has ring of integers O_{K} = ℤ[ζ_{13}] (you do not have to prove this).

(a) Factor 2, 3, 5, 7, 11, and 13 in the ring of integers O_{K}. You may use a computer.

(b) For p ≠ 13, find a conjectural relationship between the number of prime ideal factors of pO_{K} and the order of the reduction of p in (ℤ/13ℤ)^{∗}.

(c) Compute the minimal polynomial f(x) ∈ ℤ[x] of ζ_{13}. Reinterpret your conjecture as a conjecture that relates the degrees of the irreducible factors of f(x) (mod p) to the order of p modulo 13. Does your conjecture remind you of quadratic reciprocity?

5. (a) Find by hand and with proof the ring of integers of each of the following two fields: ℚ(\sqrt{5}), ℚ(i).

(b) Find the ring of integers of ℚ(x^{5} + 7x + 1) using a computer.