Let F have prime characteristic p and let a be in F. Show that the polynomial f(x)=x^p-a either splits or is irreducible in F[x].
I was given a hit: "what can you say about all of the roots of f in a splitting field?"
The Attempt at a Solution
Suppose f has a root b. That is, b^p=a. Then using Freshman's dream, we have (x-b)^p=x^p+(-b)^p=x^p-a, and so it splits. So, it suffices to show that if that polynomial is NOT irreducible, then it must have a root. That's where I got stuck. The statement seems to be false for field of zero characteristic, so the fact that Char(F)=p should be somehow used.