Proof of Fermat's Little Theorem for Prime p and Polynomial g(x) in Z_p[X]

[Dunkle]

Homework Statement

For a prime p and a polynomial g(x) that is irreducible in $$Z_{p}[X]$$, prove that for any f(x) in $$Z_{p}[X]$$ and integer k > 1, $$[f(x)]^{k} = [f(x)]$$ in $$Z_{p}[X]/(g(x))$$.

The Attempt at a Solution

I realize this is an extension of Fermat's Little Theorem, however I cannot figure out how to proceed. I attempted to adapt a proof of FLT for the integers modulo p, but couldn't get anywhere. Any nudges in the right direction would be greatly appreciated!

 

[Hurkyl]

Did you mean "there exists an integer k > 1" instead of "for every integer k > 1"?


What is the proof of FLT you're trying to adapt? What part of it doesn't work?

 

[Dunkle]

Yes, Hurkyl, I meant there exists an integer k > 1. Sorry about that.