Showing that an element of a polynomial ring is irreducible

Question

Show that $x^2 − y^3$ is irreducible in $K[x, y]$ and conclude that $K[x, y]/(x^2 − y^3$) is an integral domain.

I honestly don't know how to approach this problem I just know that i have to use the fact that $K[x, y]$ is a UFD.

Answer 1

The polynomial $x^2-y^3$ is irreducible in $(K(y))[x]$, since it is a quadric without a root over a field. Hence the polynomial is prime and the quotient is an integral domain.