Why is $y^2-x^3\in \mathbb{C}[x,y]$ irreducible? How can I prove that $y^2-x^3\in \mathbb{C}[x,y]$ is irreducible?

Grade the ring  $ \mathbb{C}[x,y]$ by $ \operatorname  {deg} x=2,\operatorname   {deg}y=3$.
A factorization  of $y^2-x^3$ would have  homogeneous factors and one of them must have degree $1,2$ or $3$.
This is impossible because the only homogeneous polynomials of these degrees are $ax$ or $by$.
