TITLE: Must $k$-subalgebra of $k[x]$ be finitely generated?
QUESTION [14 upvotes]: Suppose $k$ is a field, $A$ is a $k$-subalgebra of the polynomial ring $k[x]$. Must $A$ be a finitely generated $k$-algebra? 
Thanks.

REPLY [21 votes]: If $A=k$, nothing need to prove. We assume $f\in A$ monic non-constant, then $k[x]$ is integral over $k[f],$ thus $k[x]$ is a noetherian $k[f]$-module. It follows $A$ is also a noetherian $k[f]$-module, thus $A$ is a finitely generated $k$-algebra.
Remark. If one uses the Artin-Tate lemma we directly get $A$ is a finitely generated $k$-algebra.