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? 



about the local ring of $\mathbb{Z}_p[T]/(pT^2+T+1)$ at the prime p
That equation has a solution in $\mathbb Z\_p$. Proof: $f(a)=-pa^2-1$ 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 $\mathbb Z\_p$.


