Math 421/621

Spring 2000

Homework 6

In this problem set, are fields. denotes the automorphisms of K that fix F. Let denote a primitive 3^rd root of unity.

1. Compute the following automorphism groups:

a)

b) .

c)

d) (G)

2. With the notation as above, let , and let Show that F₁ is a subfield of K containing F and that H is a subgroup of

3. Prove the freshman’s dream holds in a field of characteristic p. That is, show that if is a field of characteristic p and , then

4. Prove Wilson’s theorem: If p is an odd prime, then

5. Find all intermediate fields between:

a) and

b) (G) and

And find all subgroups of:

c)

d)