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Spherically complete models of Hensel minimal valued fields

David Bradley-Williams, Immanuel Halupczok

Abstract

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is $0$-hmix-minimality (which, in equi-characteristic $0$, amounts to $0$-h-minimality).

Spherically complete models of Hensel minimal valued fields

Abstract

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is -hmix-minimality (which, in equi-characteristic , amounts to -h-minimality).

Paper Structure

This paper contains 4 sections, 9 theorems, 12 equations.

Table of Contents

  1. Notations and Assumptions
  2. Lemmas
  3. Spherically complete models
  4. Related questions

Key Result

Theorem 1

Let $K$ be a characteristic 0 valued field, considered as a structure in a language expanding the valued field language, such that, either Then $K$ has an elementary extension $L \succ K$ which is immediate and spherically complete (and hence maximal).

Theorems & Definitions (25)

  • Theorem 1
  • Corollary 2
  • Lemma 1.1: CHR-hmin
  • Lemma 1.2: CHRV-hmin2
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Example 1.5
  • Lemma 2.1
  • proof
  • ...and 15 more