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On algebraically closed fields with a distinguished subfield

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Abstract

This paper is concerned with the model-theoretic study of pairs (K, F) where K is an algebraically closed field and F is a distinguished subfield of K allowing extra structure. We study the basic model-theoretic properties of those pairs, such as quantifier elimination, model-completeness and saturated models. We also prove some preservation results of classification-theoretic notions such as stability, simplicity, NSOP1, and NIP. As an application, we conclude that a PAC field is NSOP1 iff its absolute Galois group is (as a profinite group).

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References

  1. H. Adler, Introduction to theories without the independence property, http://www.logic.univie.ac.at/~adler/docs/nip.pdf.

  2. B. Afshordel, Generic automorphisms with prescribed fixed fields, Journal of Symbolic Logic 79 (2014), 985–1000.

    Article  MathSciNet  Google Scholar 

  3. E. Artin and O. Schreier, Algebraische Konstruktion reeller Körper, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5 (1927), 85–99.

    Article  MathSciNet  Google Scholar 

  4. J. Ax, The elementary theory of finite fields, Annals of Mathematics 88 (1968), 239–271.

    Article  MathSciNet  Google Scholar 

  5. E. Bouscaren and B. Poizat, Des belles paires aux beaux uples, Journal of Symbolic Logic 53 (1988), 434–442.

    Article  MathSciNet  Google Scholar 

  6. I. Ben-Yaacov, A. Pillay and E. Vassiliev, Lovely pairs of models, Annals of Pure and Applied Logic 122 (2003), 235–261.

    Article  MathSciNet  Google Scholar 

  7. E. Casanovas, Simple Theories and Hyperimaginaries, Lecture Notes in Logic, Vol. 39, Association for Symbolic Logic, Chicago, IL; Cambridge University Press, Cambridge, 2011.

    Book  Google Scholar 

  8. Z. Chatzidakis, Model theory of profinite groups having the Iwasawa property, Illinois Journal of Mathematics 42 (1998), 70–96.

    Article  MathSciNet  Google Scholar 

  9. Z. Chatzidakis, Simplicity and independence for pseudo-algebraically closed fields, in Models and computability (Leeds, 1997), London Mathematical Society Lecture Note Series, Vol. 259, Cambridge University Press, Cambridge, 1999, pp. 41–61.

    Chapter  Google Scholar 

  10. Z. Chatzidakis, Properties of forking in ω-free pseudo-algebraically closed fields, Journal of Symbolic Logic 67 (2002), 957–996.

    Article  MathSciNet  Google Scholar 

  11. Z. Chatzidakis, Amalgamation of types in pseudo-algebraically closed fields and applications, Journal of Mathematical Logic 19 (2019), Article no. 1950006.

  12. C. C. Chang and H. J. Keisler, Model Theory, Studies in Logic and the Foundations of Mathematics, Vol. 73, North-Holland, Amsterdam, 1990.

    Google Scholar 

  13. A. Chernikov and N. Ramsey, On model-theoretic tree properties, Journal of Mathematical Logic 16 (2016), Article no. 1650009.

  14. A. Chernikov and P. Simon, Externally definable sets and dependent pairs. II, Transactions of the American Mathematical Society 367 (2015), 5217–5235.

    Article  MathSciNet  Google Scholar 

  15. G. Cherlin, L. van den Dries and A. Macintyre, The elementary theory of regularly closed fields, https://ms.mcmaster.ca/~cousingd/Cherlin,%20van%20den%20Dries,%20MacIntyre%20-%20The%20Elementary%20Theory%20of%20Regularly%20Closed%20Fields.pdf

  16. G. Cherlin, L. van den Dries and A. Macintyre, Decidability and undecidability theorems for PAC-fields, Bulletin of the American Mathematical Society 4 (1981), 101–104.

    Article  MathSciNet  Google Scholar 

  17. E. Casanovas and M. Ziegler, Stable theories with a new predicate, Journal of Symbolic Logic 66 (2001), 1127–1140.

    Article  MathSciNet  Google Scholar 

  18. C. d’Elbée, Forking, imaginaries and other fetures of acfg, Journal of Symbolic Logic 86 (2021), 669–700.

    Article  MathSciNet  Google Scholar 

  19. F. Delon, Élimination des quantificateurs dans les paires de corps algébriquement clos, Confluentes Mathematici 4 (2012), Article no. 1250003.

  20. M. D. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 11, Springer, Berlin, 2008

    Google Scholar 

  21. Y. Halevi and I. Kaplan, Saturated models for the working model theorist, Bulletin of Symbolic Logic 29 (2023), 163–169.

    Article  MathSciNet  Google Scholar 

  22. M. Hils, M. Kamensky and S. Rideau, Imaginaries in separably closed valued fields, Proceedings of the London Mathematical Society 116 (2018), 1457–1488.

    Article  MathSciNet  Google Scholar 

  23. W. Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, Vol. 42, Cambridge University Press, Cambridge, 1993.

    Book  Google Scholar 

  24. F. Jahnke and P. Simon, NIP henselian valued fields, Archive for Mathematical Logic 59 (2020), 167–178.

    Article  MathSciNet  Google Scholar 

  25. H. J. Keisler, Complete theories of algebraically closed fields with distinguished subfields, Michigan Mathematical Journal 11 (1964), 71–81.

    Article  MathSciNet  Google Scholar 

  26. H. J. Keisler, Ultraproducts which are not saturated, Journal of Symbolic Logic 32 (1967), 23–46.

    Article  MathSciNet  Google Scholar 

  27. B. Kim, Simplicity Theory, Oxford Logic Guides, Vol. 53, Oxford University Press, Oxford, 2014.

    Google Scholar 

  28. I. Kaplan and N. Ramsey, On Kim-independence, Journal of the European Mathematical Society 22 (2020), 1423–1474.

    Article  MathSciNet  Google Scholar 

  29. I. Kaplan and P. Simon, Witnessing dp-rank, Notre Dame Journal fo Formal Logic 55 (2014), 419–429.

    MathSciNet  Google Scholar 

  30. S. Lang, Introduction to Algebraic Geometry, Addison-Wesley, Reading, MA, 1972.

    Google Scholar 

  31. A. Macintyre, Algebra and geometry in basic model theory of fields, http://modnet.imj-prg.fr/Publications/Introductory%20Notes%20and%20surveys/macintyre.pdf.

  32. P. Morandi, Field and Galois Theory, Graduate Texts in Mathematics, Vol. 167, Springer, New York, 1996.

    Book  Google Scholar 

  33. A. Martin-Pizarro and M. Ziegler, Equational theories of fields, Journal of Symbolic Logic 85 (2020), 828–851.

    Article  MathSciNet  Google Scholar 

  34. A. Pillay, The model-theoretic content of Lang’s conjecture, in Model Theory and Algebraic Geometry, Lecture Notes in Mathematics, Vol. 1696, Springer, Berlin, 1998, pp. 101–106.

    Chapter  Google Scholar 

  35. B. Poizat, Paires de structures stables, Journal of Symbolic Logic 48 (1983), 239–249.

    Article  MathSciNet  Google Scholar 

  36. S. N. Ramsey, Independence, Amalgamation, and Trees, Ph.D. Thesis, University of California, Berkeley, CA, 2018.

    Google Scholar 

  37. A. Robinson, Solution of a problem of Tarski, Fundamenta Mathematica 47 (1959), 179–204.

    Article  MathSciNet  Google Scholar 

  38. S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, Vol. 92, North-Holland, Amsterdam, 1990.

    Google Scholar 

  39. P. Simon, A Guide to NIP Theories, Lecture Notes in Logic, Vol. 44, Association for Symbolic Logic, Chicago, IL; Cambridge Scientific Publishers, Cambridge, 2015.

    Book  Google Scholar 

  40. A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley–Los Angeles, CA, 1951.

    Book  Google Scholar 

  41. K. Tent and M. Ziegler, A Course in Model Theory, Lecture Notes in Logic, Vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012.

    Book  Google Scholar 

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Acknowledgement

The authors would like to thank Zoé Chatzidakis for her useful comments and give a special thanks to Nick Ramsey for valuable discussions and ideas in this project. We would also like to thank Anand Pillay for his comments leading us to Remark 5.15 and Question 7.3.

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Authors and Affiliations

  1. School of Mathematics, University of Leeds, Office 10.17f, LS2 9JT, Leeds, UK

    Christian d’Elbée

  2. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Itay Kaplan & Leor Neuhauser

Authors
  1. Christian d’Elbée
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  2. Itay Kaplan
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  3. Leor Neuhauser
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Corresponding author

Correspondence to Christian d’Elbée.

Additional information

The authors would like to thank the Israel Science Foundation for their support of this research (grant no. 1254/18). The first-named author was partially supported by the S.A Schonbrunn Fellowship. This paper was done as part of the third-named author’s master thesis under the supervision of the first- and second-named authors.

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d’Elbée, C., Kaplan, I. & Neuhauser, L. On algebraically closed fields with a distinguished subfield. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2621-1

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  • DOI: https://doi.org/10.1007/s11856-024-2621-1

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