Definable v -topologies, Henselianity and NIP

Yatir Halevi, Assaf Hasson, Franziska Jahnke

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3 Scopus citations

Abstract

We initiate the study of definable V-topologies and show that there is at most one such V-topology on a t-henselian NIP field. Equivalently, we show that if (K,v1,v2) is a bi-valued NIP field with v1 henselian (respectively, t-henselian), then v1 and v2 are comparable (respectively, dependent). As a consequence, Shelah's conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah's conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is t-henselian.

Original languageEnglish
Article number2050008
JournalJournal of Mathematical Logic
Volume20
Issue number2
DOIs
StatePublished - 1 Aug 2020

Keywords

  • NIP valued fields
  • Shelah conjecture
  • definable V -topologies
  • henselianity conjecture

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