# Strongly Dependent Ordered Abelian Groups and Henselian Fields

## TL;DR

This paper characterizes strongly dependent ordered abelian groups with finite dp-rank and applies this to show that strongly dependent fields with henselian valuations preserve strong dependence in their valued structures.

## Contribution

It provides a classification of strongly dependent ordered abelian groups and demonstrates that strong dependence is preserved under henselian valuations in fields.

## Key findings

- Strongly dependent ordered abelian groups have finite dp-rank.
- Such groups have finite spines and finitely many primes with infinite index.
- Strong dependence in fields extends to their henselian valued structures.

## 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 $v$.

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Source: https://tomesphere.com/paper/1706.03376