Let where is transcedental over . Let be the group of automorphism generated by the automorphism of such that . Determine and .

Suppose is the automorphism of that . Group is generated by . It is easy to verify that is the cyclic group . So If , then .

Let’s determine . Note that where is invariant under , because , for . So .

On the other hand, is a field of characteristic and is a splitting field of over . has no root in , thus it is irreducible on by a previous conclusion. So , which means where .

What if is replaced by the group of automorphism such that , ,

If , , then and

If , , then the conclusion is the same with previous problem.

If , , where .

Jacobson, Algebra I p243.