Prove that $(z-x^3,y-x^2)$ is a prime ideal

Question

I am trying to prove that the ideal in ${\mathbb C}[x,y,z]$ generated by $z-x^3$ and $y-x^2$ is prime. I know I could take a suitable quotient and show it's domain. But I am rather stuck.

Answer 1

Hint: Consider ${\mathbb C}[x,y,z] \to {\mathbb C}[x]$ induced by $x\mapsto x$, $y\mapsto x^2$, $z\mapsto x^3$.