In a book on rational series, a blunt statement is made to the effect that:
For $K$ a field, $I$ an ideal of $K[x]$, $K[x]/I$ is finitely generated iff $I$ is nonnull.
The statement elaborates with the not-so-enlightening (to me) sentence
This is true since a nonnull ideal in $K[x]$ always has a finite codimension ⁽¹⁾, and the latter is equal to the degree of any generator of this ideal ⁽²⁾.
I gather that if (2) is true, then $K[x]$ may be finitely generated if $K$ itself is finitely generated, but this is as far as I can go.
As for (1), I have a feeling of why this is true, but no proof. Thus I need help in proving the whole statement :-) Thanks !