Pseudo algebraically closed field

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

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 $f\in K[T_{1},T_{2},\cdots ,T_{r},X]$ with ${\frac {\partial f}{\partial X}}\not =0$ and for each nonzero $g\in K[T_{1},T_{2},\cdots ,T_{r}]$ there exists $({\textbf {a}},b)\in K^{r+1}$ such that $f({\textbf {a}},b)=0$ and $g({\textbf {a}})\not =0$.
• 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) 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 $e$-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". (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. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.