On algebraically closed fields with a distinguished subfield

TL;DR

This paper explores the model-theoretic properties of algebraically closed fields with a distinguished subfield, analyzing their logical structure and classification-theoretic features, with applications to PAC fields and Galois groups.

Contribution

It provides new results on quantifier elimination, model-completeness, and classification-theoretic preservation for pairs of algebraically closed fields with a subfield.

Findings

Pairs have quantifier elimination and are model-complete.

Stability, simplicity, NSOP1, and NIP are preserved under certain conditions.

PAC fields are NSOP1 iff their absolute Galois group is profinite.

Abstract

This paper is concerned with the model-theoretic study of pairs $(K,F)$ where $K$ is an algebraically closed field and $F$ is a distinguished subfield of $K$ allowing extra structure. We study the basic model-theoretic properties of those pairs, such as quantifier elimination, model-completeness and saturated models. We also prove some preservation results of classification-theoretic notions such as stability, simplicity, NSOP$_1$, and NIP. As an application, we conclude that a PAC field is NSOP$_1$ iff its absolute Galois group is (as a profinite group).