Harvard Math 129: Algebraic Number Theory Homework Assignment 10

William Stein

Due: Thursday, May 5, 2005

The problems have equal point value, and multi-part problems are of the same value. You are encouraged to look at books for inspiration about how to solve any of the following problems.

1. Let K be a number field. Prove that there is a finite set S of primes of K such

that

O_{K,S} = {a ∈ K^{∗} : ord_{𝔭}(aO_{K}) ≥ 0 all 𝔭 ∉ S} ∪ {0}

is a prinicipal ideal domain. The condition ord_{𝔭}(aO_{K}) ≥ 0 means that in the prime ideal factorization of the fractional ideal aO_{K}, we have that 𝔭 occurs to a nonnegative power. [See the notes for hints.]

2. Let a ∈ K and n a positive integer. Prove that L = K(a^{1/n}) is unramified outside the primes that divide n and the norm of a. This means that if 𝔭 is a prime of O_{K}, and 𝔭 is coprime to n Norm_{L/K}(a)O_{K}, then the prime factorization of 𝔭O_{L} involves no primes with exponent bigger than 1.

3. Write down a proof of Hilbert’s Theorem 90, formulated as the statement that for any number field K, we have

H^{1}(K, \overline{K}^{∗}) = 0.