Math 129: Algebraic Number Theory Homework Assignment 6

William Stein

Due: Thursday, March 25, 2004

1. Let p ∈ Z and let K be a number field. Show that Norm_{K/Q}(pO_{K}) = p^{[K:Q]}.

2. A totally real number field is a number field in which all embeddings into C have image in R. Prove there are totally real number fields of degree p, for every prime p. [Hint: Let ζ_{n} denote a primitive nth root of unity. For n ≥ 3, show that Q(ζ_{n} + 1/ζ_{n}) is totally real of degree φ(n)/2. Now prove that φ(n)/2 can be made divisible by any prime.]

3. Give an example of a number field K/Q and a prime p such that the e_{i} in the factorization of pO_{K} are not all the same.

4. Let K be a number field. Give the “simplest” proof you can think of that there are only finitely many primes that ramify (i.e., have some e_{i} > 1) in K. [The meaning of “simplest” is a matter of taste.]

5. Give examples to show that for K/Q a Galois extension, the quantity e can be arbirarily large and f can be arbitrarily large.

6. Suppose K/Q is Galois and p is a prime such that pO_{K} is also prime (i.e., p is inert in K). Show that Gal(K/Q) is a cyclic group.

7. (Problem 7, page 116, from Marcus Number Fields) For each of the following, find a prime p and quadratic extensions K and L of Q that illustrates the assertion:

(a) The prime p can be totally ramified in K and L without being totally ramified in KL.

(b) The fields K and L can each contain unique primes lying over p while KL does not.

(c) The prime p can be inert in K and L without being inert in KL.

(d) The residue field extensions of F_{p} can be trivial for K and L without being trivial for KL.