What is the Krull dimension of $\mathbb{Q}[x_1,x_2,x_3]/(x_1^3 + x_2x_3^2)$

Question

I am having some difficulty computing the Krull dimension of $\mathbb{Q}[x_1,x_2,x_3]/(x_1^3 + x_2x_3^2)$.

Could anybody give me some suggestions? Thank you!

Answer 1

$\mathbf Q[x_1,x_2,x_3]$ is a regular ring of dimension $3$. As it is a finitely generated algebra over a field and $\;x_1^3+x_2x_3^2$ is a non-zero divisor, Krull's Hauptidealsatz ensures $$\dim\bigl(\mathbf Q[x_1,x_2,x_3]/(x_1^3+x_2x_3^2)\bigr)=\dim\mathbf Q[x_1,x_2,x_3] -1=2$$