$(X^2-Y^3)$ is prime ideal in $K[X,Y]$

Question

Let $K$ be a field. I want to show that $(X^2-Y^3)\subset K[X,Y]$ is a prime ideal.

I know that $K[X,Y]$ is factorial, so it is enough to show that $X^2-Y^3$ is irreducible. But Eisenstein does not to work here, does it?

How can I see that $(X^2-Y^3)$ is prime ideal?

Thanks in advance.

Edit: The task is to show that $K[X,Y]/(X^2-Y^3)$ is a domain. I thought it is best to show that $(X^2-Y^3)$ is a prime ideal in $K[X,Y]$. Am I mistaken?

Answer 1

Hint: Prove that the homomorphism $K[X,Y] \to K[T]$ given by $p(X,Y) \mapsto p(t^3,t^2)$ has kernel $(X^2-Y^3)$.

Then, $K[X,Y]/(X^2-Y^3)$ is isomorphic to a subring of $K[T]$, which is a domain, and so is itself a domain.

Finally, recall that $I$ is a prime ideal of a commutative ring $R$ iff $R/I$ is a domain.