How to prove that the algebra $k[x, xy, xy^2, xy^3, \dotsc]$ is not finitely generated?
Consider the subalgebra $R=k[x,xy,xy^2,xy^3, \ldots] \subset k[x,y]$. How do I prove that $R$ is not finitely generated (over $k$)?
What is the general strategy for proving that an algebra is not finitely generated?
Suppose that $R$ is generated by $f_1,\dots,f_n$ over $k$, which are polynomials in terms of $xy^m$ up to $m = M$. ($M$ is finite since there are finitely many $f_1,\dots,f_n$). This would imply that $R = k + (x,xy,\dots,xy^M)$. Now we can go back to your previous question to see that for example, $xy^{M+1}$ does not lie in $(x,xy,\dots,xy^M)$.