On Pseudo Algebraically Closed Extensions of Fields

TL;DR

This paper develops a framework for studying Pseudo Algebraically Closed (PAC) extensions of fields, proving that the Galois closure of proper separable algebraic PAC extensions is their separable closure, and classifying finite PAC extensions.

Contribution

It introduces a generalized machinery based on embedding problems for field extensions, proving new properties and classifications of PAC extensions, and unifying existing results.

Findings

Galois closure of proper separable algebraic PAC extensions equals their separable closure

Classification of all finite PAC extensions

Proof of the bottom conjecture for finitely generated infinite fields

Abstract

The notion of `Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert's tenth problem, and recently had new applications. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure. This vastly generalizes earlier works of Jarden-Razon, Jarden, and Jarden and the author. This also leads to a classification of all finite PAC extensions which in turn proves the `bottom conjecture' for finitely generated infinite fields. The secondary goal of this work is to unify proofs of known results about PAC extensions and to establish new basic properties of PAC extensions, e.g.\ transitiveness of PAC extensions.