Still in my "commutative algebra marathon", I came across the following exercise:
Any $k$-subalgebra $A$ of $k[x]$ is finitely generated as $k$-algebra; also, if $A\ne k$, then $\dim A=1$.
Although I have found a related answer here, I don't get how to follow the steps mentioned there. Thus, any help to understand the linked answer above is welcome $-$ or a new answer to the original problem.
Thanks.