Is an integral domain $\frac{\mathbb{C}[x,y]}{<x^4+x^3y+y^4>}$?

Question

Is an integral domain $\frac{\mathbb{C}[x,y]}{<x^4+x^3y+y^4>}$ ?
Where $\mathbb{C}[x,y]$ is a commutative ring of polynomials over $\mathbb{C}.$

I know the fact that for any field, $\mathbb F[x,y]/ \left\langle xy+b \right\rangle $ is an integral domain iff $b\neq 0$.

Answer 1

Over the complex numbers one has $$x^4+x^3y+y^4=(x-\alpha_1y)(x-\alpha_2y)(x-\alpha_3y)(x-\alpha_4y)$$ where $\alpha_1,\ldots,\alpha_4$ are the roots of $$z^4+z^3+1=0.$$ Then each $x-\alpha_jy$ is a divisor of zero in $R=\Bbb C[x,y]/\langle x^4+x^3y+y^4\rangle$, so $R$ is not an integral domain.