Pseudo algebraically closed field

In mathematics, a field ${\displaystyle K}$ 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.[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 ${\displaystyle e}$-tuple ${\displaystyle (\sigma _{1},...,\sigma _{e})\in G^{e}}$, 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

1. Fried & Jarden (2008) p.218
2. Fried & Jarden (2008) p.192
3. Fried & Jarden (2008) p.449
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

• 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. Zbl 1145.12001.