Formal Derivative and Multiple roots

Question

I am currently really stuck on the following problem:

Prove that if f(x) in Fp[x] and Df = 0 (where D : Fp[x] → Fp[x] is the formal derivative) then there exists g(x) in Fp[x] such that f(x) = g(x)^p

I know that there are multiple roots, since hcf(f, Df) is non-constant, but I have no idea how to show all these roots are of the same polynomial g(x).

Any tips would be appreciated. Thanks

Answer 1

If the formal derivative is the zero polynomial, then all the terms in $f$ must have exponents that are multiples of $p$. Then $(a+b)^p=a^p+b^p$ comes to the rescue.