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Definable henselian valuations on dp-minimal real fields

Lothar Sebastian Krapp, Salma Kuhlmann, Lasse Vogel

Abstract

We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.

Definable henselian valuations on dp-minimal real fields

Abstract

We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.

Paper Structure

This paper contains 3 sections, 6 theorems, 8 equations.

Table of Contents

  1. Definable valuations on almost real closed fields
  2. Classification of the definable valuations of a dp-minimal real field
  3. Further work

Key Result

Proposition 1.5

Let $K$ be an almost real closed field and let $p$ be a prime. Consider the $\mathcal{L}_r$-formulae Then $\varphi_p(K) := \mathcal{O}_{v_p}$, i.e. the valuation $v_p$ is definable with defining formula $\varphi_p(x)$.

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Proposition 1.5
  • proof
  • Lemma 1.7
  • proof
  • Theorem 1.8
  • proof
  • Remark 1.9
  • ...and 9 more