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.
F[x]/(fg) is isomorphic with F[x]/(f) x F[x]/(g).
Any help greatly appreciated.