Pseudo algebraically closed field

Formulation

• 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 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 -tuple, 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".
• Let K be the maximal totally real Galois extension of the rational numbers and i the square root of -1. Then K is PAC.

  Properties

• 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.