is the localisation of the ring $$A:=\mathbb{Z}_p[T]/(pT^2+T+1)$$ at the prime ideal (p) isomorphic to $\mathbb{Z}_p$?
If not, how to understand this ring very explicitly?
is the localisation of the ring $$A:=\mathbb{Z}_p[T]/(pT^2+T+1)$$ at the prime ideal (p) isomorphic to $\mathbb{Z}_p$? If not, how to understand this ring very explicitly? 


That equation has a solution in $\mathbb Z_p$. Proof: $f(a)=pa^21$ is a contraction mapping for the $p$adic metric, and a fixed point must be a solution. Let $\alpha$ be that solution, then $(T\alpha)$ divides your equation. The other factor must be $pT\alpha^{1}$. So we have: $A = \mathbb Z_p[T]/(T\alpha) \times \mathbb Z_p [T]/(pT\alpha^{1})=\mathbb Z_p \times \mathbb Q_p$ The localization at $p$ is $\mathbb Z_p$. 

