Henselian Valued Fields Research Articles

A transfer theorem for Henselian valued and ordered fields

AbstractIn well-known papers ([A-K1], [A-K2], and [E]) J. Ax, S. Kochen, and J. Ershov prove a transfer theorem for henselian valued fields. Here we prove an analogue for henselian valued and ordered fields. The orders for which this result apply are the usual orders and also the higher level orders introduced by E. Becker in [Bl] and [B2]. With certain restrictions, two henselian valued and ordered fields are elementarily equivalent if and only if their value groups (with a little bit more structure) and their residually ordered residue fields (a henselian valued and ordered field induces in a natural way an order in its residue field) are elementarily equivalent. Similar results are proved for elementary embeddings and ∀-extensions (extensions where the structure is existentially closed).

Indécidabilité de la théorie des paires immédiates de corps valués henseliens

AbstractThe theory of immediate pairs of Henselian valued fields, with a given residual theory (of characteristic zero) and a given theory of valuation group (nonzero), is undecidable and has completions.

Relative elimination of quantifiers for Henselian valued fields

Basarab, S.A., Relative elimination of quantifiers for Henselian valued fields, Annals of Pure and Applied Logic 53 (1991) 51–74 A general result on relative elimination of quantifiers for Henselian valued fields of characteristic zero is proved by algebraic and basic model-theoretic methods.

On the angular component map modulo P

In [10] we introduced a new first order language for valued fields. This language has three sorts of variables, namely variables for elements of the valued field, variables for elements of the residue field and variables for elements of the value group. contains symbols for the standard field, residue field, and value group operations and a function symbol for the valuation. Essential in our language is a function symbol for an angular component map modulo P, which is a map from the field to the residue field (see Definition 1.2).For this language we proved a quantifier elimination theorem for Henselian valued fields of equicharacteristic zero which possess such an angular component map modulo P [10, Theorem 4.1]. In the first section of this paper we give some partial results on the existence of an angular component map modulo P on an arbitrary valued field.By applying the above quantifier elimination theorem to ultraproducts ΠQp/D, we obtained a quantifier elimination, in the language , for the p-adic field Qp; and this elimination is uniform for almost all primes p [10, Corollary 4.3]. In §2 we prove that our language is essentially stronger than the natural language for p-adic fields in the sense that the angular component map modulo P cannot be defined, uniformly for almost all p, in terms of the natural language for p-adic fields.

Local differents of algebraic and finite extensions of valued fields

The properties of discriminants and differents were studied first by Dedekind and Hilbert in finite algebraic extensions of fields of algebraic numbers. From a local point of view, that is equivalent to a study of the p-adic case, where the results of Dedekind and Hilbert can be formulated as follows. Dedekind's theorem: The g.c.d. Δ( K k ) of differents of integral bases of a finite algebraic extension K k (which I call an algebraic different if K k and the g.c.d δ( K k ) of differents of integral elements of K k (which I call an arithmetic different of K k ) coincide; Hilbert's theorem (which is the basis of Herbrand's ramification theory of intermediate extensions): If K ⊃ L ⊃ k, δ( K k ) = δ( K L ) δ( L k ) . These results are easily generalizable to the “classical case” of henselian valued basic fields, i.e., the case when the valuation is discrete and the residual extension K k of K k is separable. But, in the general case of extensions K k of valued fields (where k may be assumed to be henselian), Dedekind's and Hilbert's theorems are not always true: the algebraic different Δ( K k ) divides the arithmetic different δ( K k ) , but generally δ( K k ) ≠ Δ( K k ) , and Hilbert's theorem holds only for the algebraic different. When the valuation is discrete, I call an extension K k dedekindian when δ( K k ) = Δ( K k ) and hilbertian if, for every intermediate field L of K k (i.e., K ⊇ L ⊇ k), Hilbert's theorem δ( K k ) = δ( K L ) δ( L k ) for arithmetic differents holds. When the valuation is dense, the situation is more complicated, because of the existence of two kinds of ideals (principal and other), and it is convenient to define dedekindian and hilbertian extensions in a slightly different manner and to introduce somewhat wider classes of extensions called quasi-dedekindian and quasi-hilbertian. I study the relations between Δ( K k ) and δ( K k ) , and, in particular, I give a complete characterization of dedekindian extensions for both discrete and dense valuations; I also give examples of non-dedekindian and non-hilbertian extensions. In Section 4, some connections with the ramification theory (both for normal and non-normal extensions) are studied and a weak analog of Hilbert's theorem [ δ( K k ) δ( L k ) divides δ( K k ) ] is proved.

Corps et anneaux de Rolle

Brown, Craven and Pelling have proved that if the polynomials over an ordered field K K satisfy Rolle’s theorem, they satisfy it for any ordering on K K . We say that such a field is a Rolle field. We prove that this is a first order property in the language of rings, and that the theory of Rolle fields is decidable. Then we give a common generalisation of these fields and the real closed rings defined by Cherlin and Dickmann: The polynomials over an ordered ring A A satisfy Rolle’s theorem iff A A is the valuation ring of a henselian valued field with Rolle residue field and m m -divisible value group for all odd m m .

A model-theoretic transfer theorem for henselian valued fields.

A model-theoretic transfer theorem for henselian valued fields.

Some model theory for Henselian valued fields

Some model theory for Henselian valued fields

Henselian Valued Fields With Prescribed Value Group and Residue Field

A class of fields supplementary to the inertia field of a given henselian valued field is used to construct extensions of that field having prescribed value group and residue field. The extensions so-constructed are characterized, and their number investigated.