scholarly journals Transfer Principles in Henselian Valued Fields

2021 ◽  
Vol 27 (2) ◽  
pp. 222-223
Author(s):  
Pierre Touchard
Keyword(s):  
Term Structure ◽  
Residue Field ◽  
Intermediate Structure ◽  
Elementary Extension ◽  
Similar Reduction ◽  
Valued Fields ◽  
Valued Field ◽  
Henselian Valued Fields ◽  
Leading Term ◽  
Transfer Principles

AbstractIn this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$ , algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$ is inp-minimal (of burden 1), and that the ring of Witt vectors $W(\mathbb {F}_p^{\text {alg}})$ over $\mathbb {F}_p^{\text {alg}}$ is not strong (of burden $\omega $ ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R}((t))$ are definable. Similarly, all types over the quotient field of $W(\mathbb {F}_p^{\text {alg}})$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye.These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups.Abstract prepared by Pierre Touchard.E-mail: [email protected]: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9

scholarly journals DEFINABLE HENSELIAN VALUATIONS

2015 ◽  
Vol 80 (1) ◽  
pp. 85-99 ◽  
Author(s):  
FRANZISKA JAHNKE ◽  
JOCHEN KOENIGSMANN
Keyword(s):  
Galois Group ◽  
Residue Field ◽  
Absolute Galois Group ◽  
Valued Fields ◽  
Valued Field ◽  
Henselian Valued Fields ◽  
The Absolute ◽  
Henselian Valuation

AbstractIn this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal.

On the angular component map modulo P

1990 ◽  
Vol 55 (3) ◽  
pp. 1125-1129 ◽  
Author(s):  
Johan Pas
Keyword(s):  
Natural Language ◽  
Residue Field ◽  
Quantifier Elimination ◽  
Function Symbol ◽  
Valued Fields ◽  
Valued Field ◽  
Angular Component ◽  
Henselian Valued Fields ◽  
Almost All ◽  
Component Map

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.

On Crossed Product Algebras over Henselian Valued Fields

2020 ◽  
Vol 27 (03) ◽  
pp. 389-404
Author(s):  
Driss Bennis ◽  
Karim Mounirh
Keyword(s):  
Positive Integer ◽  
Division Algebra ◽  
Residue Field ◽  
Crossed Product ◽  
Valued Fields ◽  
Central Division ◽  
Valued Field ◽  
Central Division Algebra ◽  
Henselian Valued Fields ◽  
Maximal Subfield

Let D be a tame central division algebra over a Henselian valued field E, [Formula: see text] be the residue division algebra of D, [Formula: see text] be the residue field of E, and n be a positive integer. We prove that Mn([Formula: see text]) has a strictly maximal subfield which is Galois (resp., abelian) over [Formula: see text] if and only if Mn(D) has a strictly maximal subfield K which is Galois (resp., abelian) and tame over E with ΓK ⊆ ΓD, where ΓK and ΓD are the value groups of K and D, respectively. This partially generalizes the result proved by Hanke et al. in 2016 for the case n = 1.

scholarly journals NOTES ON EXTREMAL AND TAME VALUED FIELDS

2016 ◽  
Vol 81 (2) ◽  
pp. 400-416
Author(s):  
SYLVY ANSCOMBE ◽  
FRANZ-VIKTOR KUHLMANN
Keyword(s):  
Approximation Property ◽  
Elementary Theory ◽  
Residue Field ◽  
Axiom System ◽  
Complete Characterization ◽  
Valued Fields ◽  
Mixed Characteristic ◽  
Valued Field ◽  
Residue Fields

AbstractWe extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

scholarly journals ON THE MAIN INVARIANT OF ELEMENTS ALGEBRAIC OVER A HENSELIAN VALUED FIELD

2002 ◽  
Vol 45 (1) ◽  
pp. 219-227 ◽  
Author(s):  
Kamal Aghigh ◽  
Sudesh K. Khanduja
Keyword(s):  
Upper Bound ◽  
Algebraic Closure ◽  
Subject Classification ◽  
Unique Extension ◽  
Valued Fields ◽  
Valued Field ◽  
Henselian Valued Fields ◽  
Alpha Beta ◽  
Henselian Valuation

AbstractLet $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

scholarly journals Valued fields with finitely many defect extensions of prime degree

2021 ◽  
pp. 2250049
Author(s):  
Franz-Viktor Kuhlmann
Keyword(s):  
Open Problem ◽  
Positive Characteristic ◽  
Residue Field ◽  
Prime Degree ◽  
Valued Fields ◽  
Mixed Characteristic ◽  
Valued Field ◽  
Field Of Positive Characteristic

We prove that a valued field of positive characteristic [Formula: see text] that has only finitely many distinct Artin–Schreier extensions (which is a property of infinite NTP2 fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has [Formula: see text]-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin–Schreier defect extensions.

A transfer theorem for Henselian valued and ordered fields

1993 ◽  
Vol 58 (3) ◽  
pp. 915-930 ◽  
Author(s):  
Rafel Farré
Keyword(s):  
Residue Field ◽  
Valued Fields ◽  
Transfer Theorem ◽  
Ordered Field ◽  
Existentially Closed ◽  
Henselian Valued Fields ◽  
Ordered Fields ◽  
Elementary Embeddings ◽  
Residue Fields ◽  
Natural Way

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).

Grothendieck Rings of ℤ-Valued Fields

2001 ◽  
Vol 7 (2) ◽  
pp. 262-269 ◽  
Author(s):  
Raf Cluckers ◽  
Deirdre Haskell
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Local Fields ◽  
Grothendieck Ring ◽  
Valued Fields ◽  
Logical Language ◽  
Valued Field ◽  
Grothendieck Rings

AbstractWe prove the triviality of the Grothendieck ring of a ℤ-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K2 to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

Valued Fields

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Field Theory ◽  
Model Theory ◽  
Newton Diagram ◽  
Basic Field ◽  
Valued Fields ◽  
Basic Properties ◽  
Basic Model ◽  
Valued Field ◽  
Henselian Valued Fields ◽  
Basic Facts

This chapter introduces the reader to basic field theory by focusing on valued fields. It first considers valuations on fields before discussing the basic properties of valued fields, with emphasis on extensions. It then describes pseudoconvergence in valued fields, along with henselian valued fields. It also shows how to decompose a valuation on a field into simpler ones, leading to an analysis of various special types of pseudocauchy sequences. Because the valuation of is compatible with its natural ordering, some basic facts about fields with compatible ordering and valuation are presented. The chapter concludes by reviewing some basic model theory of valued fields as well as the Newton diagram and Newton tree of a polynomial over a valued field.

scholarly journals Relative decidability and definability in henselian valued fields

2011 ◽  
Vol 76 (4) ◽  
pp. 1240-1260 ◽  
Author(s):  
Joseph Flenner
Keyword(s):  
Decision Procedure ◽  
Quantifier Elimination ◽  
Main Step ◽  
Linear Map ◽  
Definable Set ◽  
Valued Field ◽  
Definable Subset ◽  
Henselian Valued Fields ◽  
Leading Term ◽  
Definable Function

AbstractLet (K, v) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial f(x) ∈ K[x], what is v(f(x))?Two applications are given: first, a constructive quantifier elimination relative to the leading terms, suggesting a relative decision procedure; second, a presentation of every definable subset of K as the pullback of a definable set in the leading terms subjected to a linear translation.

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