I have a proof of the Chinese Remainder Theorem for integers right here, but I'm not sure how to prove it for polynomials with the following criteria:

Let F be a field. Let f(x) and g(x) be two non-constant polynomials such that gcd(f,g)=1. They are relatively prime.

Prove that:

F[x]/(fg) is isomorphic with F[x]/(f) x F[x]/(g).

Any help greatly appreciated.