On pseudo algebraically closed extensions of fields

Abstract

The notion of ‘Pseudo Algebraically Closed (PAC) extensions’ is a generalization of the classical notion of PAC fields. 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. As a result we get 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.