Let $Y=\operatorname{Spec}\Bbb Z[T]/(T^2+1)$ and $X=\operatorname{Spec}\Bbb Z$. Prove that X and Y are regular.
My attempt:
Denote $\Bbb Z_{p}$ the localisation of $\Bbb Z$ at $p$.
- X is regular:
We know that a PID is regular and that localization of regular ring is regular, so we can immediately state that $\Bbb Z$ is regular and that $\forall p \in Spec\ \Bbb Z: \Bbb Z_{p}$ is regular. Hence $X$ is regular.
But let me try it by hand: $\forall \mathfrak p \in Spec\ \Bbb Z:\Bbb Z_{p}$ is a local ring of maximal ideal $\mathfrak m:=p\Bbb Z_{\mathfrak p}$. Then:
$\dim_{\Bbb F_p} \mathfrak m/\mathfrak m^2 = \dim_{\Bbb F_p} p\Bbb Z/p^2\Bbb Z = 1=\dim_{\Bbb F_p}\Bbb Z_{p}$ hence $\Bbb Z_{p}$ is regular and $X$ is regular.
- Y is regular:
I don't know where to start, computations is complicated here and I would need a helping hand.
Thank you.