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

Question

Given the quotient ring $R:=\mathbb{C}[x,y,z]/(xy-z^2)$, I've to check whether it's a field or an integral domain.

I'd say that $R$ is a field and an integral domain because $(xy-z^²)$ is irreducible in $\mathbb{C}[x,y,z]$ (Eisenstein with $x \in \mathbb{C}[x,y][z]$). However, I know that $R$ is not a field. I don't know what I've done wrong. Is my application of Eisenstein criteria false? How can I show that $R$ is not a field, but an integral domain?

Answer 1

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?

Answer 2

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 $1$?