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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

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1 Answer 1

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.

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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
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