Rahul Sharma (Editor)

Pseudo algebraically closed field

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

Contents

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 ∈ K [ T 1 , T 2 , ⋯ , T r , X ] with ∂ f ∂ X ≠ 0 and for each nonzero g ∈ K [ T 1 , T 2 , ⋯ , T r ] there exists ( a , b ) ∈ K r + 1 such that f ( a , b ) = 0 and g ( a ) ≠ 0 .
  • Each absolutely irreducible polynomial f ∈ 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 → K such that h ( a ) = a for each a ∈ 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 -tuple ( σ 1 , . . . , σ e ) ∈ 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.

    • References

    Pseudo algebraically closed field Wikipedia
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