Groups and fields with 𝑁𝑇𝑃₂

Chernikov, Artem; Kaplan, Itay; Simon, Pierre

doi:10.1090/S0002-9939-2014-12229-5

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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