David Marker, Introduction to the model theory of fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 1–37. - David Marker. Model theory of differential fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 38–113. - Anand Pillay. Differential algebraic groups and the number of countable differentially closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 114–134. - Margit Messmer. Some model theory of separably closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 135–152.

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.

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Diophantine Geometry from Model Theory

Bulletin of Symbolic Logic ◽

10.2307/2687822 ◽

2001 ◽

Vol 7 (1) ◽

pp. 37-57 ◽

Cited By ~ 2

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.

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Existentially closed fields with finite group actions

Journal of Mathematical Logic ◽

10.1142/s0219061318500034 ◽

2018 ◽

Vol 18 (01) ◽

pp. 1850003 ◽

Cited By ~ 2

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.

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Minimal types in separably closed field

Journal of Symbolic Logic ◽

10.2307/2586707 ◽

2000 ◽

Vol 65 (3) ◽

pp. 1443-1450 ◽

Cited By ~ 4

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.

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THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

Journal of Symbolic Logic ◽

10.1017/jsl.2015.76 ◽

2016 ◽

Vol 81 (2) ◽

pp. 493-509

Author(s):

OMAR LEÓN SÁNCHEZ ◽

RAHIM MOOSA

Keyword(s):

Mathematical Logic ◽

Characteristic Zero ◽

Differential Algebra ◽

Model Companion ◽

Finite Algebras ◽

Closed Fields ◽

Hensel’S Lemma ◽

Differential Fields ◽

Theory Of Fields ◽

Model Theory Of Fields

AbstractA model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.

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Model theory of fields with free operators in characteristic zero

Journal of Mathematical Logic ◽

10.1142/s0219061314500093 ◽

2014 ◽

Vol 14 (02) ◽

pp. 1450009 ◽

Cited By ~ 8

Author(s):

Rahim Moosa ◽

Thomas Scanlon

Keyword(s):

Free Algebra ◽

Model Theory ◽

Characteristic Zero ◽

Model Companion ◽

Finite Dimensional ◽

Differential Fields ◽

Theory Of Fields ◽

Model Theory Of Fields

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme 𝒟 over a field A of characteristic zero, the theory of 𝒟-fields has a model companion 𝒟-CF0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.

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