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Strongly Dependent Ordered Abelian Groups and Henselian Fields

Yatir Halevi, Assaf Hasson

Abstract

Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and $|\{p\text{ prime}:[G:pG]=\infty\}|<\infty$. We apply this to show that if $K$ is a strongly dependent field, then $(K,v)$ is strongly dependent for any henselian valuation $v$.

Strongly Dependent Ordered Abelian Groups and Henselian Fields

Abstract

Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and . We apply this to show that if is a strongly dependent field, then is strongly dependent for any henselian valuation .

Paper Structure

This paper contains 8 sections, 34 theorems, 74 equations.

Table of Contents

  1. introduction
  2. Preliminaries and notation
  3. Ordered abelian groups
  4. Examples
  5. Strong dependence, burden and dp-rank
  6. Ordered abelian groups with finite spines
  7. Calculating the dp-rank
  8. Strongly dependent henselian fields

Key Result

Theorem 1

Let $G$ be an ordered abelian group. The following are equivalent

Theorems & Definitions (90)

  • Theorem 1
  • Theorem 2
  • Remark
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Remark
  • Definition 3.1
  • Remark
  • ...and 80 more