p-Convexly valued rings

Abstract

In (J. Symbolic Logic 56(2) (1991) 539), Bélair developed a theory analogous to the theory of real closed rings in the p-adic context, namely the theory of p-adically closed integral rings. Firstly we use the property proved in Lemma 2.4 in (J. Symbolic Logic 60(2) (1995) 484) to express this theory in a language including a p-adic divisibility relation and we show that this theory admits definable Skolem functions in this language (in the sense of (J. Symbolic Logic 49 (1984) 625)). Secondly, we are interested in dealing with some questions similar to that of (Z. Math. Logik Grundlag. 29(5) (1983) 417); e.g., results about integral-definite polynomials over a p-adically closed integral ring A and a kind of "Nullstellensatz" using the notion of M A -radical.

Henselian residually p-adically closed fields

“…So, by Lemma 2.2, we get that v p ( p.x) v p (y), which implies v p (x) < v p (y). Now we recall some definitions and results from [7], namely the notions of p-valued and p-convexly valued domains. It is useful in the next theorems for the following reasons:…”

“…relation with respect to a p-valuation v p and D v as a l.d. relation with respect to a valuation v. The L p -theory of p-convexly valued domains is denoted by pC V R. An axiomatization of pC V R in L p can be found in Section 2 of [7]. Now we recall a part of Lemma 2.9 in [7].…”

“…See Lemma 2.14 in [7] where pcH (A, K ) is defined as follows Then there exists a p-convexly valued domain A such that…”

“…See Lemma 2.15 in [7]. Now we recall the definition of the Kochen's operator which plays an important role in the characterization of p-valued field extensions (see Chapter 6 in [13]).…”

“…We introduce the field analogue of the notion of M-Kochen ring which was considered in Section 3 of [7] for valued domains. It allows us to characterize the intersection of the valuation rings of p-valuations which extend a fixed p-valuation v p such that v p (M) 0 for some particular subset M. Since we want to use a model completeness result, we have to identify the subset M which is required in the solution of this problem for henselian residually p-adically closed fields.…”