Cross-Sections for p-Adically Closed Fields

HENSELIAN RESIDUALLY p-ADICALLY CLOSED FIELDS

In (Arch. Math. 57 (1991), pp. 446–455), R. Farré proved a positivstellensatz for real-series closed fields. Here we consider p-valued fields 〈K, vp〉 with a non-trivial valuation v which satisfies a compatibility condition between vp and v. We use this notion to establish the p-adic analogue of real-series closed fields; these fields are called henselian residually p-adically closed fields. Fir...

The Elementary Theory of Free Pseudo p-adically Closed Fields of Finite Corank

Cell decomposition for P-minimal fields

In [S-vdD] P. Scowcroft and L. van den Dries prove a Cell Decomposition Theorem for p-adically closed fields. We work here with the notion of P -minimal fields defined by D. Haskell and D. Macpherson in [H-Mph]. We prove that a P -minimal field K admits cell decomposition if and only if K has definable selection. A preprint version in French of this result appeared as a prepublication [M].

Computable p–adic Numbers

In the present work the notion of the computable (primitive recursive, polynomially time computable) p–adic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are p–adically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterizatio...

Elementary equivalence versus Isomorphism

1) The isomorphy type of a finite field K is given by its cardinality |K|, i.e., if K and L are such fields, then K ∼= L iff |K| = |L|. 1) The isomorphy type of an algebraically closed field K is determined by two invariants: (i) Absolute transcendence degree td(K), (ii) The characteristic p = char(K) ≥ 0. In other words, if K and L are algebraically closed fields, then K ∼= L iff td(K) = td(L)...