Field of characteristic p. automosphism.

Let be a field of characteristic . Let be an extension of . Define .

a) Prove that is a subfield of .

b) Show that any automorphism of leaving every element of fixed also leaves every element of fixed.

ATTEMPT:

Part (a) is easy after observing that .

Now part (b). Let be an automorphism with .

NOTATION: and so on.

Now consider the special case when . We need to show that .

Since we have .

Thus . This leads to and also to the conclusion that .

Consider .

If these are all distinct then the polynomial will have distinct roots. Since this is impossible thus some two of

the elements are same. This leads to the conclusion that such that .

Now we need to show that the minimum value of such an is one.

How do I proceed from here?