# Quotient Ring Proof

1. Nov 18, 2009

### Dunkle

1. The problem statement, all variables and given/known data

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 $$Z_{p}[X]/(g(x))$$.

3. 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!

2. Nov 18, 2009

### Hurkyl

Staff Emeritus
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?

3. Nov 18, 2009

### Dunkle

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