Pseudo algebraically closed field

In mathematics, a field [math]\displaystyle{ K }[/math] 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.

Formulation

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

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

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 [math]\displaystyle{ G }[/math] of a field [math]\displaystyle{ K }[/math] is profinite, hence compact, and hence equipped with a normalized Haar measure. Let [math]\displaystyle{ K }[/math] be a countable Hilbertian field and let [math]\displaystyle{ e }[/math] be a positive integer. Then for almost all [math]\displaystyle{ e }[/math]-tuple [math]\displaystyle{ (\sigma_1,...,\sigma_e)\in G^e }[/math], 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

• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9.