Let $\varphi : F[X] \rightarrow R'$ be a ring homomorphism where $F$ is a field and $R'$ is an integral domain. $P = \ker \varphi$ is either maximal or $(0)$.
I know that the maximal ideals of $F[X]$ correspond to the principal ideals generated by irreducible monic polynomials, and that $P$ is maximal iff $F[X]/P$ is a a field. Please only give a hint. Thanks.