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EQUATIONAL THEORIES OF FIELDS

Part of: Model theory Differential and difference algebra

Published online by Cambridge University Press:  15 July 2020

AMADOR MARTIN-PIZARRO  and
MARTIN ZIEGLER
AMADOR MARTIN-PIZARRO
Affiliation:
MATHEMATISCHES INSTITUT ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGD-79104FREIBURG, GERMANYE-mail: pizarro@math.uni-freiburg.deE-mail: ziegler@uni-freiburg.de
MARTIN ZIEGLER
Affiliation:
MATHEMATISCHES INSTITUT ALBERT-LUDWIGS-UNIVERSITÄT FREIBURGD-79104FREIBURG, GERMANYE-mail: pizarro@math.uni-freiburg.deE-mail: ziegler@uni-freiburg.de
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Abstract

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.

Keywords

model theoryseparably closed fieldsdifferentially closed fieldsequationality

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 12H05: Differential algebra
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 828 - 851
Copyright
© The Association for Symbolic Logic 2020

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References

REFERENCES

Ben-Yaacov, I., Pillay, A., and Vassiliev, E., Lovely pairs of models . Annals of Pure and Applied Logic, vol. 122 (2003), pp. 235–261.CrossRefGoogle Scholar
Delon, F., Idéaux et types sur les corps séparablement clos . Les Mémoires de la Société Mathématique de France, vol. 33 (1988), p. 76.Google Scholar
Delon, F., Élimination des quantificateurs dans les paires de corps algébriquement clos . Confluentes Mathematici, vol. 4 (2012), p. 11.CrossRefGoogle Scholar
Dieudonné, J., Cours de géométrie algébrique, Le Mathématicien, Presses Universitaires de France, Paris, 1974, p. 222.Google Scholar
Günaydın, A., Topological study of pairs of algebraically closed fields, preprint, 2017. Available from https://arxiv.org/pdf/1706.02157.pdf Google Scholar
Junker, M., A note on equational theories , this Journal, vol. 65 (2000), pp. 1705–1712.Google Scholar
Junker, M. and Lascar, D., The indiscernible topology: A mock Zariski topology . Journal of Mathematical Logic, vol. 1 (2001), pp. 99–124.CrossRefGoogle Scholar
Keisler, H. J., Complete theories of algebraically closed fields with distinguished subfields . Michigan Mathematical Journal, vol. 11 (1964), pp. 71–81.CrossRefGoogle Scholar
Lang, S., Algebra, second ed., Addison-Wesley Publishing Company, Boston, MA, 1984.Google Scholar
O’Hara, A., An introduction to equations and equational theories, preprint, 2011. Available from http://www.math.uwaterloo.ca/rmoosa/ohara.pdf Google Scholar
Martin-Pizarro, A. and Ziegler, M., Equational theories of fields: An extended version, preprint, 2017. Available from https://arxiv.org/abs/1702.05735 Google Scholar
Müller, I. and Sklinos, R., Nonequational stable groups, preprint, 2017. Available from https://arxiv.org/abs/1703.04169 Google Scholar
Pillay, A., Imaginaries in pairs of algebraically closed fields . Annals of Pure and Applied Logic, vol. 146 (2007), pp. 13–20.CrossRefGoogle Scholar
Pillay, A. and Srour, G., Closed sets and chain conditions in stable theories , this Journal, vol. 49 (1984), pp. 1350–1362.Google Scholar
Poizat, B., Paires de structures stables , this Journal, vol. 48 (1983), pp. 239–249.Google Scholar
Sela, Z., Free and hyperbolic groups are not equational, preprint, 2013. Available from http://www.ma.huji.ac.il/zlil/equational.pdf.Google Scholar
Srour, G., The independence relation in separably closed fields , this Journal, vol. 51 (1986), pp. 751–725.Google Scholar
Srour, G., The notion of independence in categories of algebraic structures, Part I: Basic properties . Annals of Pure and Applied Logic, vol. 38 (1988), pp. 185–213.CrossRefGoogle Scholar