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 3rd root of unity.

 

  1. Compute the following automorphism groups:

 

a)

b) .

c)

d) (G)

 

  1. 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

 

  1. 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

 

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

 

  1. Find all intermediate fields between:

 

a)       and

b)      (G)  and

 And find all subgroups of:

c)     

d)