XVIIème problème de Hilbert sur les corps chaîne-clos

1991 ◽  
Vol 56 (3) ◽  
pp. 853-861
Author(s):  
Françoise Delon et Danielle Gondard
Keyword(s):  
Residue Field ◽  
Algebraic Extension ◽  
Real Field ◽  
Closed Field ◽  
Several Variables ◽  
A Chain ◽  
Henselian Valuation

AbstractA chain-closed field is defined as a chainable field (i.e. a real field such that, for all n ∈ N, ΣK2n+2 ≠ ΣK2n) which does not admit any “faithful” algebraic extension, and can also be seen as a field having a Henselian valuation ν such that the residue field K/ν is real closed and the value group νK is odd divisible with ∣νK/2νK∣ = 2. If K admits only one such valuation, we show that f ∈ K(X) is in ΣK(X)2n for any real algebraic extension L of K,“f(L) ⊆ ΣL2n” holds. The conclusion is also true for K = R((t))(a chainable but not chain-closed field), and in the case n = 1 it holds for several variables and any real field K.

Constructing complete distinguished chains with given invariants

2014 ◽  
Vol 14 (03) ◽  
pp. 1550026
Author(s):  
Kamal Aghigh ◽  
Azadeh Nikseresht
Keyword(s):  
Algebraic Closure ◽  
Residue Field ◽  
Finite Sequence ◽  
Unique Extension ◽  
Arbitrary Rank ◽  
A Chain ◽  
Henselian Valuation

Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and [Formula: see text] be the unique extension of v to a fixed algebraic closure [Formula: see text] of K with value group [Formula: see text]. It is known that a complete distinguished chain for an element θ belonging to [Formula: see text] with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of [Formula: see text], a tower of fields, together with a sequence of elements belonging to [Formula: see text] which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of [Formula: see text] containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of [Formula: see text] satisfying certain properties, it is shown that there exists a complete distinguished chain for an element [Formula: see text] associated to these invariants. We use the notion of lifting of polynomials to construct it.

Some model theory for almost real closed fields

1996 ◽  
Vol 61 (4) ◽  
pp. 1121-1152 ◽  
Author(s):  
Françoise Delon ◽  
Rafel Farré
Keyword(s):  
Model Theory ◽  
Residue Field ◽  
Elementary Equivalence ◽  
First Order ◽  
Valuation Rings ◽  
Closed Fields ◽  
Theory Of Fields ◽  
Model Theory Of Fields ◽  
Elementary Inclusion ◽  
Henselian Valuation

AbstractWe study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

The model theory of ordered differential fields

1978 ◽  
Vol 43 (1) ◽  
pp. 82-91 ◽  
Author(s):  
Michael F. Singer
Keyword(s):  
Meromorphic Functions ◽  
Analytic Solutions ◽  
Positive Element ◽  
Real Closed Field ◽  
Real Field ◽  
Closed Field ◽  
Ordered Field ◽  
Positive Elements ◽  
Odd Degree ◽  
Differential Fields

In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.§ 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.

Going-Down Results for Ci-Fields

2006 ◽  
Vol 49 (1) ◽  
pp. 11-20
Author(s):  
Anthony J. Bevelacqua ◽  
Mark J. Motley
Keyword(s):  
Function Fields ◽  
Algebraic Extension ◽  
Real Closed Field ◽  
Closed Field ◽  
Difficult Case ◽  
Lang's Conjecture ◽  
Lang’S Conjecture

AbstractWe search for theorems that, given a Ci-field K and a subfield k of K, allow us to conclude that k is a Cj -field for some j. We give appropriate theorems in the case K = k(t) and K = k((t)). We then consider the more difficult case where K/k is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field.

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.

scholarly journals Levels of division algebras

1990 ◽  
Vol 32 (3) ◽  
pp. 365-370 ◽  
Author(s):  
David B. Leep
Keyword(s):  
Function Field ◽  
Algebraic Extension ◽  
Global Field ◽  
Real Closed Field ◽  
Closed Field ◽  
Division Algebras ◽  
Central Division ◽  
Product Level ◽  
Euclidean Field ◽  
Finite Dimensional

In [7] the level, sublevel, and product level of finite dimensional central division algebras D over a field F were calculated when F is a local or global field. In Theorem 1.4 of this paper we calculate the same quantities if all finite extensions K of F satisfy ū(K) ≤2, where ū is the Hasse number of a field as defined in [2]. This occurs, for example, if F is an algebraic extension of the function field R(x) where R is a real closed field or hereditarily Euclidean field (see [4]).

Quantifier elimination in valued Ore modules

2010 ◽  
Vol 75 (3) ◽  
pp. 1007-1034 ◽  
Author(s):  
Luc Bélair ◽  
Françoise Point
Keyword(s):  
Residue Field ◽  
Quantifier Elimination ◽  
Difference Operators ◽  
Independence Property ◽  
Valued Fields ◽  
A Chain

AbstractWe consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.

PETERZIL–STEINHORN SUBGROUPS AND -STABILIZERS IN ACF

2021 ◽  
pp. 1-20
Author(s):  
Moshe Kamensky ◽  
Sergei Starchenko ◽  
Jinhe Ye
Keyword(s):  
Algebraic Group ◽  
Residue Field ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Valued Field

Abstract We consider G, a linear algebraic group defined over $\Bbbk $ , an algebraically closed field (ACF). By considering $\Bbbk $ as an embedded residue field of an algebraically closed valued field K, we can associate to it a compact G-space $S^\mu _G(\Bbbk )$ consisting of $\mu $ -types on G. We show that for each $p_\mu \in S^\mu _G(\Bbbk )$ , $\mathrm {Stab}^\mu (p)=\mathrm {Stab}\left (p_\mu \right )$ is a solvable infinite algebraic group when $p_\mu $ is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of $\mathrm {Stab}\left (p_\mu \right )$ in terms of the dimension of p.

COMPARING TWO VERSIONS OF THE REALS

2016 ◽  
Vol 81 (3) ◽  
pp. 1115-1123
Author(s):  
G. IGUSA ◽  
J. F. KNIGHT
Keyword(s):  
Residue Field ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Computing Power ◽  
Ordered Field ◽  
Image Position ◽  
Recursively Saturated

AbstractSchweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$. We prove that ${\cal R}$ is not reducible to ${\cal W}$. As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.

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