Dunkle
Nov18-09, 12:10 AM
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!
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!