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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Groups and fields with $\operatorname {NTP}_{2}$
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by Artem Chernikov, Itay Kaplan and Pierre Simon PDF
Proc. Amer. Math. Soc. 143 (2015), 395-406 Request permission

Abstract:

$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in $\operatorname {NTP}_{2}$ structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any $\operatorname {NTP}_{2}$ field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.
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Additional Information
  • Artem Chernikov
  • Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel
  • Address at time of publication: L’équipe de Logique Mathématique, IMJ-PRG, Université Paris Diderot-Paris 7, UFR de Mathématiques, case 7012, 75205 Paris Cedex 13, France
  • Email: art.chernikov@gmail.com
  • Itay Kaplan
  • Affiliation: Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
  • Address at time of publication: Institute of Mathematics, Hebrew University (The Edmond J. Safra Campus), Givat Ram, Jerusalem 91904, Israel
  • MR Author ID: 886730
  • Email: itay.kaplan@uni-muenster.de
  • Pierre Simon
  • Affiliation: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel
  • Address at time of publication: Université Claude Bernard-Lyon 1, Institut Camille Jordan, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
  • MR Author ID: 942320
  • Email: pierre.simon@normalesup.org
  • Received by editor(s): December 31, 2012
  • Received by editor(s) in revised form: February 26, 2013
  • Published electronically: August 19, 2014
  • Additional Notes: The first author was partially supported by the [European Community’s] Seventh Framework Programme [FP7/2007-2013] under grant agreement No. 238381
    The second author was supported by SFB 878
  • Communicated by: Julia Knight
  • © Copyright 2014 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 395-406
  • MSC (2010): Primary 03C45, 03C60
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12229-5
  • MathSciNet review: 3272764