I want to show that $K(X,Y)$ is not finitely generated $K$ algebra($K$ is a field) where $X ,Y$ are algebraically independent over $K$.
I proved it using Noether Normalisation as following. If it was finitely generated $K$ algebra then there was $X_1, \ldots,X_m$ algebraically independent over $K$ such that $K(X,Y)$ were integral over $K[X_1,\ldots,X_m].$ Then $m=0$ since $dim(K(X,Y))=0.$ Thus $K(X,Y)$ becomes algebraic over $K,$ which is not true since $X$ is not algebraic over $K.$
But I want to prove it in the sprit of the problem that $\mathbb{Q}$ is not finitely generated $\mathbb{Z}$ algebra as there are infinitely many non associate primes in $\mathbb{Z}.$ If there is any solution involving the number of primes then it will be helpful for me. Though I know that there are infinitely many primes in the polynomial ring $K[X,Y].$
Thank you.