(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?"

2. Relevant equations

3. 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.

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# Homework Help: Char(F)=p; poly NOT irreduc. => must have a root?

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