Check whether the quotient ring $R:=\mathbb{C}[x,y,z]/(xy-z^2)$ is a field or an integral domain

Hint:

$\mathbf C[X,Y,Z]$ is a U.F.D., so ideals generated by irreducible elements have height $1$. Can these ideals be maximal?

A quotient ring is a field if and only if the original ideal is maximal. Use the fact that $\mathbb{C}[x,y,z]$ has natural gradings by degree; does $(xy-z^2)$ contain any elements of degree $1$?