Krull dimension of a $\mathbb Q$-algebra

Question

I'm trying to find the Krull dimension of $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$.

My professor said that I have to consider that $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})$ is a $\mathbb{Q}$-algebra, but I don't know how to proceed.

I think I have to find a transcendence basis for this $\mathbb{Q}$-algebra, but I don't how to make the computations.

Any help will be welcome. Thank you.

Answer 1

You can write $\mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2})=\mathbb Q[x,y,z]$ with $x^2=y$ and $z^2=0$. Thus you have an integral extension $\mathbb Q[y]\subset\mathbb Q[x,y,z]$. Now try to show that $\mathbb Q[y]\simeq\mathbb Q[Y]$ (that is, $y$ is transcendental over $\mathbb Q$) and find out the Krull dimension.