# Integral extension inside a polynomial ring over a field

Let $K$ be a field and $D = K[X]$. I need to show that if $f\in D$ is non constant, then the extension of rings $K[f]\subset D$ is integral, and if $A$ is a subring of $D$ which contains $K$ and has Krull dimension $1$, then $A$ is a finitely generated $K$-algebra.

• What have you tried so far? You can assume wlog that $f$ is monic. Now you can write down explicitly a polynomial which shows that $X$ is integral over $K[f]$. etc. – Martin Brandenburg Jun 18 '13 at 13:31