Pseudo algebraically closed field

In mathematics, a field $$K$$ is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.

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


 * Each absolutely irreducible variety $$V$$ defined over $$K$$ has a $$K$$-rational point.
 * For each absolutely irreducible polynomial $$f\in K[T_1,T_2,\cdots ,T_r,X]$$ with $$\frac{\partial f}{\partial X}\not =0$$ and for each nonzero $$g\in K[T_1,T_2,\cdots ,T_r]$$ there exists $$(\textbf{a},b)\in K^{r+1}$$ such that $$f(\textbf{a},b)=0$$ and $$g(\textbf{a})\not =0$$.
 * Each absolutely irreducible polynomial $$f\in K[T,X]$$ has infinitely many $$K$$-rational points.
 * If $$R$$ is a finitely generated integral domain over $$K$$ with quotient field which is regular over $$K$$, then there exist a homomorphism $$h:R\to K$$ such that $$h(a) = a$$ for each $$a \in K$$.

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 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 $$G$$ of a field $$K$$ is profinite, hence compact, and hence equipped with a normalized Haar measure. Let $$K$$ be a countable Hilbertian field and let $$e$$ be a positive integer. Then for almost all $$e$$-tuples $$(\sigma_1,...,\sigma_e)\in G^e$$, 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.

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.