Formulation

• Each absolutely irreducible
  Absolutely irreducible
  In mathematics, absolutely irreducible is a term applied to linear representations or algebraic varieties over a field. It means that the object in question remains irreducible, even after any finite extension of the field of coefficients...
  
   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.

• A non-principal ultraproduct
  Ultraproduct
  The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...
  
   of distinct finite fields is PAC (consequence of Riemann Hypothesis for function fields).

• Infinite algebraic extension
  Algebraic extension
  In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
  
  s of finite field
  Finite field
  In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
  
  s are PAC (consequence of Riemann Hypothesis for function fields).

• This example arises from measure theory: The absolute Galois group
  Absolute Galois group
  In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
  
    of a field is profinite, hence compact
  Compact group
  In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
  
  , and hence equipped with a normalized Haar measure
  Haar measure
  In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
  
  . 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". (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.