In mathematics, a field is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.Fried & Jarden (2008) p.218

Formulation

A field K is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds:

Each absolutely irreducible variety defined over has a -rational point. For each absolutely irreducible polynomial with and for each nonzero there exists such that and . Each absolutely irreducible polynomial has infinitely many -rational points. If is a finitely generated integral domain over with quotient field which is regular over , then there exist a homomorphism such that for each .

Examples

Algebraically closed fields and separably closed fields are always PAC. Pseudo-finite fields and hyper-finite fields are PAC. A non-principal ultraproduct of distinct finite fields is (pseudo-finite and henceFried & Jarden (2008) p.449) PAC.Fried & Jarden (2008) p.192 Ax deduces this from the Riemann hypothesis for curves over finite fields. Infinite algebraic extensions of finite fields are PAC.Fried & Jarden (2008) p.196 The PAC Nullstellensatz. The absolute Galois group of a field is profinite, hence compact, and hence equipped with a normalized Haar measure. Let be a countable Hilbertian field and let be a positive integer. Then for almost all -tuples , the fixed field of the subgroup generated by the automorphisms is PAC. Here the phrase "almost all" means "all but a set of measure zero".Fried & Jarden (2008) p.380 (This result is a consequence of Hilbert's irreducibility theorem.) Let K be the maximal totally real Galois extension of the rational numbers and i the square root of −1. Then K(i) is PAC.

Properties The Brauer group of a PAC field is trivial,Fried & Jarden (2008) p.209 as any Severi–Brauer variety has a rational point. The absolute Galois group of a PAC field is a projective profinite group; equivalently, it has cohomological dimension at most 1.Fried & Jarden (2008) p.210 A PAC field of characteristic zero is C1.Fried & Jarden (2008) p.462

References