Pseudo algebraically closed field explained

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.^[1]

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, …} ,T _{r,X]} with (\partial f)/(\partial X) \not =0 and for each nonzero g\in K[T _{1,T} _{2, …} ,T _{r]} there exists ( bf{a},b)\in K ^{r+1} such that f( bf{a},b)=0 and g( bf{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^[2]) PAC.^[3] Ax deduces this from the Riemann hypothesis for curves over finite fields.^[1]
• Infinite algebraic extensions of finite fields are PAC.^[4]

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".^[5] (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,^[6] 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.^[7]
• A PAC field of characteristic zero is C1.^[8]

References

• Book: Fried . Michael D. . Jarden . Moshe . Field arithmetic . 3rd revised . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge . 11 . . 2008 . 978-3-540-77269-9 . 1145.12001 .

Notes and References

1. Fried & Jarden (2008) p.218
2. Fried & Jarden (2008) p.449
3. Fried & Jarden (2008) p.192
4. Fried & Jarden (2008) p.196
5. Fried & Jarden (2008) p.380
6. Fried & Jarden (2008) p.209
7. Fried & Jarden (2008) p.210
8. Fried & Jarden (2008) p.462