Pseudo algebraically closed field

In mathematics, a field ${\displaystyle K}$ 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^[2]) if one of the following equivalent conditions holds:

• Each absolutely irreducible variety ${\displaystyle V}$ defined over ${\displaystyle K}$ has a ${\displaystyle K}$-rational point.
• For each absolutely irreducible polynomial ${\displaystyle f\in K[T_{1},T_{2},\cdots ,T_{r},X]}$ with ${\displaystyle {\frac {\partial f}{\partial X}}\not =0}$ and for each nonzero ${\displaystyle g\in K[T_{1},T_{2},\cdots ,T_{r}]}$ there exists ${\displaystyle ({\textbf {a}},b)\in K^{r+1}}$ such that ${\displaystyle f({\textbf {a}},b)=0}$ and ${\displaystyle g({\textbf {a}})\not =0}$.
• Each absolutely irreducible polynomial ${\displaystyle f\in K[T,X]}$ has infinitely many ${\displaystyle K}$-rational points.
• If ${\displaystyle R}$ is a finitely generated integral domain over ${\displaystyle K}$ with quotient field which is regular over ${\displaystyle K}$, then there exist a homomorphism ${\displaystyle h:R\to K}$ such that ${\displaystyle h(a)=a}$ for each ${\displaystyle a\in K}$.

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^[3]) PAC.^[2] Ax deduces this from the Riemann hypothesis for curves over finite fields.^[1]
• Infinite algebraic extensions of finite fields are PAC.^[4]
• The PAC Nullstellensatz. The absolute Galois group ${\displaystyle G}$ of a field ${\displaystyle K}$ is profinite, hence compact, and hence equipped with a normalized Haar measure. Let ${\displaystyle K}$ be a countable Hilbertian field and let ${\displaystyle e}$ be a positive integer. Then for almost all ${\displaystyle e}$-tuples ${\displaystyle (\sigma _{1},...,\sigma _{e})\in G^{e}}$, 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.^[7]
• 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

• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.