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Characterizing NIP henselian fields

Sylvy Anscombe, Franziska Jahnke

Abstract

In this paper, we characterize NIP henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.

Characterizing NIP henselian fields

Abstract

In this paper, we characterize NIP henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.

Paper Structure

This paper contains 7 sections, 25 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic notions from valuation theory
  3. NIP valued fields
  4. NIP transfer from residue field to valued field
  5. The main theorem and immediate consequences
  6. A model-theoretic version of Theorem \ref{['thm:1']}
  7. A refinement of Conjecture \ref{['sh']}

Key Result

Lemma 2.4

There is an $\mathcal{L}_{\mathrm{val}}$-theory $T_{d}$ which axiomatizes the class of defectless valued fields.

Figures (1)

  • Figure 1: The Standard Decomposition

Theorems & Definitions (60)

  • Conjecture 1.1: Conjecture on NIP fields
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 50 more