[Math] $\phi: F[x] \rightarrow R$ is a ring homomorphism to an integral domain. Why must ker$(\phi)$ be a maximal ideal or (0)

ring-theory

Here is my working so far:

The kernel of any ring homomorphism is an ideal.

If $\phi$ is injective then ker$(\phi)$ is (0).

I don't know what to do if $\phi$ isn't injective. I know that the maximal ideals of $F[x]$ are generated by monic irreducible polynomials. I was thinking something along the lines of assuming that the kernel is generated by two polynomials, and arguing by contradiction.

Best Answer

  1. Make use of a fundamental theorems of ring homomorphisms: $\mathrm{Im\,}\phi \cong F[x]/\ker \phi$.

  2. Note that the image is an integral domain (being a subring of one).

  3. Think about for which type of ideals $R/I$ can be integral domain.

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