Abstract Algebra

[dabien]

1. The problem statement, all variables and given/known data
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).


2. Relevant equations



3. 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.