Is $x^2y^2 - z^3$ an irreducible element in $\Bbb Q[x,y,z]$?
I want to prove $\Bbb Q[x,y,z]/(x^2y^2-z^3)$ is an integral domain, so I need to show $(x^2y^2 - z^3)$ is a prime ideal of $\Bbb Q[x,y,z]$. I tried to show $x^2y^2-x^3$ is an irreducible element, and I tried to apply Eisenstein’s criterion of prime ideals, but each variable $x,y,z$ has more degree greater than $2$. Therefore I need to use another method, but nothing comes upon to me.
Think of $z^3-x^2y^2$ as a monic cubic in $z$. If it had a factor, it would have a linear factor of the form $z-\phi$ where $\phi\in\Bbb Q[x,y]$ and $\phi^3=x^2y^2$. There is no such $\phi$.