Groups and fields with NTP₂

NTP₂ is a large class of first-order theories defined by Shelah 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.