On Pseudo Algebraically Closed Extensions of Fields

Details On Pseudo Algebraically Closed Extensions of Fields On Pseudo Algebraically Closed Extensions of Fields

2008-06-24

arxiv.org/abs/0806.3851v3

Mathematics

Number Theory

31 pages, small corrections, change of commutative diagrams, to be published in the Journal of Algebra

Scientific paper

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.

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