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.
Thank you in any advance.