Given a field F show that every proper non trivial prime ide

Given a field F show that every proper non trivial prime ideal of F[x] is maximal.

Solution

Let I be a nontrivial prime ideal of F[x]. Since I is a principal ideal generated by f for some f F[x]. We know that I is maximal if and only if f is irreducible. We need to show that if I is prime, then f is irreducible. However, if I is prime, then if f factors as f = gh, we must have g I or h I. Since f generates I, this means that g = f k or h = f k. This forces g or h to be constant polynomials, so f is irreducible.