Kernel of ring homomorphism from a polynomial ring over field to a field is maximal ideal or zero ideal

Question

Question If $E,F$ are fields and $\beta:F[x]\rightarrow E$ a homomorphism of rings. Show that the kernel of $\beta$ is a maximal ideal or a zero ideal.

I just wonder where does the zero ideal case arise? I haven't seen any necessity to discuss the situation of zero ideal. Any help would be appreciated!

Answer 1

Let $E$ be the field of fractions of $F[x]$ and $\beta$ the obvious embedding.