Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
up vote 4 down vote accepted

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.

share|cite|improve this answer
Thank you so much! It's clearly getting late and I'm not thinking too straight. Thanks again. B – Mt123 Feb 11 '13 at 1:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.