Groups and fields with NTP2

TL;DR

This paper explores the properties of groups and fields within NTP2 theories, revealing chain conditions, finiteness of Artin-Schreier extensions, and characteristics of strongly dependent valued fields.

Contribution

It introduces new chain conditions for definable subgroups in NTP2 structures and establishes finiteness results for Artin-Schreier extensions in NTP2 fields.

Findings

NTP2 fields have finitely many Artin-Schreier extensions

A chain condition for definable normal subgroups in NTP2 structures

Strongly dependent valued fields are Kaplansky

Abstract

NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 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 NTP2 structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any NTP2 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.