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.

scholarly journals Existentially closed fields with finite group actions

2018 ◽  
Vol 18 (01) ◽  
pp. 1850003 ◽  
Author(s):  
Daniel M. Hoffmann ◽  
Piotr Kowalski
Keyword(s):  
Finite Group ◽  
Model Theory ◽  
Group Scheme ◽  
General Context ◽  
Finite Group Actions ◽  
Existentially Closed ◽  
Closed Fields ◽  
A Finite Group ◽  
Theory Of Fields ◽  
Model Theory Of Fields

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.

Minimal types in separably closed field

2000 ◽  
Vol 65 (3) ◽  
pp. 1443-1450 ◽  
Author(s):  
Zoé Chatzidakis ◽  
Carol Wood
Keyword(s):  
Model Theory ◽  
Abelian Varieties ◽  
Transcendence Degree ◽  
Closed Field ◽  
Closed Fields ◽  
Theory Of Fields ◽  
Model Theory Of Fields

In [1], examples of types of U-rank 1 (i.e., minimal types) in the theories of separably closed fields were constructed, en route to displaying certain dimension phenomena. We construct here additional examples with U-rank 1 and of various transcendence degrees over arbitrary separably closed fields. Our examples include ones which are minimal but of infinite transcendence degree, i.e., not thin. Our interest in building new examples was piqued after seeing the role played by minimal types over separably closed fields in Hrushovski's analysis of abelian varieties. This article is the result of several working sessions between the authors at Wesleyan University and Paris 7, and was completed during the Model Theory of Fields program at MSRI in 1998. We are grateful for the hospitality and support of all three institutions. We thank Elisabeth Bouscaren and Françoise Delon for reading an earlier version of this paper, providing useful suggestions and corrections.

Model Theory of Fields

2016 ◽  
Author(s):  
David Marker ◽  
Margit Messmer ◽  
Anand Pillay
Keyword(s):  
Model Theory ◽  
Theory Of Fields ◽  
Model Theory Of Fields

Model Theory of Fields

2005 ◽  
Author(s):  
David Marker ◽  
Margit Messmer ◽  
Anand Pillay
Keyword(s):  
Model Theory ◽  
Theory Of Fields ◽  
Model Theory Of Fields

Introduction to the Model Theory of Fields

2017 ◽  
pp. 1-37 ◽  
Author(s):  
David Marker ◽  
David Marker ◽  
Margit Messmer ◽  
Anand Pillay
Keyword(s):  
Model Theory ◽  
Theory Of Fields ◽  
Model Theory Of Fields

Diophantine Geometry from Model Theory

2001 ◽  
Vol 7 (1) ◽  
pp. 37-57 ◽  
Author(s):  
Thomas Scanlon
Keyword(s):  
Model Theory ◽  
Stability Theory ◽  
Function Field ◽  
Algebraic Groups ◽  
Diophantine Geometry ◽  
Diophantine Problems ◽  
Theory Of Fields ◽  
Model Theory Of Fields

Abstract§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery could unlock the secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields.

Anneaux de fonctions p-adiques

1995 ◽  
Vol 60 (2) ◽  
pp. 484-497 ◽  
Author(s):  
Luc Bélair
Keyword(s):  
Prime Ideal ◽  
Residue Field ◽  
Local Rings ◽  
First Order ◽  
Valuation Rings ◽  
Integrally Closed ◽  
Definable Functions ◽  
Quotient Rings

AbstractWe study first-order properties of the quotient rings (V)/ by a prime ideal where (V) is the ring of p-adic valued continuous definable functions on some affine p-adic variety V. We show that they are integrally closed Henselian local rings, with a p-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy ∀ x, y(x∣y2 ∨ y∣x2).

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