Abstract Algebra: Proving E=F(a^p)

[dabien]

Homework Statement

Let F be a field of characteristic p>0 and let E = F(a) where a is separable over F. Prove that E=F(a^p).

Homework Equations

The Attempt at a Solution

I know that maybe show how mod F(a) = mod F(a^p) or something around there.

 

[icantadd]

So, since the characteristic is p, a prime E is generated by some field F adjoined some set of elements, and for each minimal polynomial associated to these elements there are no duplicate roots.

Now, there are two cases to consider E is finite or E is infinite. If E is finite then you're done, since E is iso to Z mod p^k. If E is infinite (for example Z4(u) where u is some transcendental element) then you have some work to do. Consider what frobenius endomorphism
http://en.wikipedia.org/wiki/Frobenius_endomorphism
Tells about the minimal polynomial of a.

 

[Hurkyl]

E is iso to Z mod p^k.

No it's not. Such an isomorphism can only exist when k=1 and E is a field whose cardinality is p.