Is $(p)$, $p$ prime, ideal prime in $\mathbb{Z}[x,y]$?

Question

Is correct the following? I want verify if $(p)$ is prime in $\mathbb{Z}[x,y]$ ($p$ prime) I have this: $\mathbb{Z}[x,y]/(p)\simeq (\mathbb{Z}[x]/(p))[y]\simeq (\mathbb{Z}_{p}[x])[y]$ and $\mathbb{Z}_{p}$ is a field then $(\mathbb{Z}_{p}[x])$ is a Euclidean domain. From here I don't know what more information to get

Answer 1

As you have shown that $$\frac{\Bbb Z[x,y]}{(p)} \cong (\Bbb Z_p[x])[y]$$

Now $\Bbb Z_p \text{ is a field } \implies \Bbb Z_p[x] \text{ is a PID } \implies (\Bbb Z_p[x])[y] \text{ is an Integral Domain} \implies (p) \in \operatorname{Spec(\Bbb Z[x,y])}$