The Stacks project

Exercise 111.55.3. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring whose residue field has characteristic not $2$. Suppose that $\mathfrak m$ is generated by three elements $x, y, z$ and that $x^2 + y^2 + z^2 = 0$ in $A$.

1. What are the possible values of $\dim (A)$?

2. Give an example to show that each value is possible.

3. Show that $A$ is a domain if $\dim (A) = 2$. (Hint: look at $\bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}$.)