Pseudo algebraically closed field

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. 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 . 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 hence) PAC. Ax deduces this from the Riemann hypothesis for curves over finite fields. Infinite algebraic extensions of finite fields are PAC. 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". (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. The Brauer group of a PAC field is trivial, 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. A PAC field of characteristic zero is C1.