Strongly minimal reducts of ACVF

TL;DR

This paper proves that certain strongly minimal, non-locally modular structures definable in algebraically closed valued fields necessarily interpret an infinite field, revealing deep connections between model-theoretic properties and algebraic structures.

Contribution

It establishes that strongly minimal, non-locally modular reducts of algebraically closed valued fields interpret an infinite field, advancing understanding of their algebraic and model-theoretic complexity.

Findings

Strongly minimal non-locally modular structures interpret infinite fields.

Reducts of algebraically closed valued fields have rich algebraic interpretations.

The work extends known results in model theory of valued fields.

Abstract

In this document we prove: Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be an algebraically closed valued field and let $(G,\oplus)$ be a $\mathbb K$-definable group that is either the multiplicative group or contains a finite index subgroup that is $\mathbb K$-definably isomorphic to a $\mathbb K$-definable subgroup of $(K,+)$. Then if $\mathcal G=(G,\oplus,\ldots)$ is a strongly minimal non locally modular structure definable in $\mathbb K$ and expanding $(G,\oplus)$, it interprets an infinite field. This document is the PhD thesis of the author and it was advised by professors Assaf Hasson and Alf Onshuus.