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: 



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



Algebraic $p$-adic integers mod $p$
This is an extended comment.


Let $\Gamma=\mathrm{Gal}(\overline{\mathbf{Q}}\_p|\mathbf{Q}\_p)$. Your question pertains to the filtration $V\subset T\subset\Gamma$, where $T$ is the inertia subgroup and $V$ the ramification subgroup, so that $\overline{\mathbf{Q}}\_p^T$ is the maximal unramified extension of $\mathbf{Q}\_p$ and $\overline{\mathbf{Q}}\_p^V$ is the maximal tamely ramified extension of $\mathbf{Q}\_p$. The quotient $\Gamma/T$ is $\hat{\mathbf{Z}}$, and the quotient $T/V$ can be identified with the group of roots of $1$ of order prime to $p$.


But the filtration (in the upper numbering) on $\Gamma$ is much finer, and it is indexed by the reals $\geqslant-1$; we have $\Gamma^0=T$ and $\Gamma^{0+}=V$. This filtration is separated in the sense that the intersection of all $\Gamma^u$ (for $u\geqslant-1$) is trivial, so $\overline{\mathbf{Q}}\_p$ is the inductive limit of the various $K^u=\overline{\mathbf{Q}}\_p^{\Gamma^u}$. As a result the $\overline{\mathbf{F}}\_p$-algebra $\overline{\mathbf{Z}}\_p/p\overline{\mathbf{Z}}\_p$ is the inductive limit (increasing union) of $\mathfrak{o}^u/p\mathfrak{o}^u$, where $\mathfrak{o}^u$ is the ring of integers of $K^u$, and $u>0$


It might be interesting to be able to say something sensible about each individual quotient $\mathfrak{o}^u/p\mathfrak{o}^u$ (and also about the quotients corresponding to $\Gamma^{u+}$, the union of all $\Gamma^v$ such that $v>u$).


