In studying the papers of Abbes and Saito on ramification theory in the imperfect residue field case, I come to the following questions. Let $\overline{\mathbb{Q}}_p\supset\mathbb{Q}{}^{t}_p\supset \mathbb{Q}{}^{nr}_p$ be an algebraic closure of the field $\mathbb{Q}_p$ of $p$-adics with its maximal tamely ramified and unramified subextensions. Let $\overline{\mathbb{Z}}_p\supset\mathbb{Z}{}^{t}_p\supset \mathbb{Z}{}^{nr}_p$ be the rings of integers of these fields. My question is:

What do the quotient rings $\overline{\mathbb{Z}}_p/(p)$, $\mathbb{Z}{}^{t}_p/(p)$ and $\mathbb{Z}{}^{nr}_p/(p)$ look like?

In fact I'm interested in all quotients $\overline{\mathbb{Z}}_p/(\pi_a)$, $\mathbb{Z}{}^{t}_p/(\pi_a)$ and $\mathbb{Z}{}^{nr}_p/(\pi_a)$ for an element $\pi_a$ of given rational valuation $a$, but I guess that the mod $p$ quotients are more tractable and give a first idea.

`$\bar{\mathbb{Z}}_p/(p) \simeq \mathcal{O}_{\mathbb{C}_p}/(p) \simeq \mathcal{O}_{\mathbb{C}_p}/(\pi)$`

for every $\pi$ with $p$-adic valuation in $p^\mathbb{Z}$, the first iso induced by completion and the second one by powers of Frobenius. It's zero-dimensional, local, non-Noetherian, the maximal ideal is idempotent and nil, the residue field is $\bar{\mathbb{F}}_p$. – Torsten Schoeneberg Apr 5 '13 at 12:53`$\mathbb{Z}_p^t/(p)$`

is local with nilpotent maximal ideal, but I couldn't go further and I guess that one can't say much more than that. If the comments are turned into an answer, I'll accept it. – Matthieu Romagny Apr 5 '13 at 18:18