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The | In [[mathematics]], a [[field (mathematics)|field]] <math>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.<ref name=MF218>Fried & Jarden (2008) p.218</ref> | ||
==Formulation== | |||
A field ''K'' is pseudo algebraically closed (usually abbreviated by '''PAC'''<ref name=MF192/>) if one of the following equivalent conditions holds: | |||
*Each [[absolutely irreducible]] variety <math>V</math> defined over <math>K</math> has a <math>K</math>-[[rational point]]. | |||
*For each absolutely irreducible polynomial <math>f\in K[T_1,T_2,\cdots ,T_r,X]</math> with <math>\frac{\partial f}{\partial X}\not =0</math> and for each nonzero <math>g\in K[T_1,T_2,\cdots ,T_r]</math> there exists <math>(\textbf{a},b)\in K^{r+1}</math> such that <math>f(\textbf{a},b)=0</math> and <math>g(\textbf{a})\not =0</math>. | |||
*Each absolutely irreducible polynomial <math>f\in K[T,X]</math> has infinitely many <math>K</math>-rational points. | |||
*If <math>R</math> is a finitely generated [[integral domain]] over <math>K</math> with [[quotient field]] which is [[regular extension|regular]] over <math>K</math>, then there exist a homomorphism <math>h:R\to K</math> such that <math>h(a)=a</math> for each <math>a\in K</math> | |||
== Examples == | |||
* Algebraically closed fields and [[separably closed]] fields are always PAC. | |||
*[[Pseudo-finite field]]s and [[hyper-finite field]]s are PAC. | |||
* A non-principal [[ultraproduct]] of distinct finite fields is (pseudo-finite and hence<ref name=FJ449>Fried & Jarden (2008) p.449</ref>) PAC.<ref name=MF192>Fried & Jarden (2008) p.192</ref> Ax deduces this from the [[Riemann hypothesis for curves over finite fields]].<ref name=MF218/> | |||
* Infinite [[algebraic extension]]s of [[finite field]]s are PAC.<ref name=MF196>Fried & Jarden (2008) p.196</ref> | |||
* ''The PAC Nullstellensatz''. The [[absolute Galois group]] <math>G</math> of a field <math>K</math> is [[profinite]], hence [[compact group|compact]], and hence equipped with a normalized [[Haar measure]]. Let <math>K</math> be a countable [[Hilbertian field]] and let <math>e</math> be a positive integer. Then for almost all <math>e</math>-tuple <math>(\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".<ref name=MF380>Fried & Jarden (2008) p.380</ref> (This result is a consequence of Hilbert's irreducibility theorem.) | |||
* Let ''K'' be the maximal [[totally real field|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,<ref name=MF209>Fried & Jarden (2008) p.209</ref> as any [[Severi–Brauer variety]] has a rational point.<ref name=MF210/> | |||
* The [[absolute Galois group]] of a PAC field is a [[projective profinite group]]; equivalently, it has [[cohomological dimension]] at most 1.<ref name=MF210>Fried & Jarden (2008) p.210</ref> | |||
* A PAC field of [[Characteristic (algebra)|characteristic]] zero is [[C1 field|C1]].<ref name=MF462>Fried & Jarden (2008) p.462</ref> | |||
== References == | |||
{{reflist}} | |||
* {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 }} | |||
[[Category:Algebraic geometry]] | |||
[[Category:Field theory]] | |||
Revision as of 20:08, 30 May 2013
In mathematics, a field 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 defined over has a -rational point.
- For each absolutely irreducible polynomial with and for each nonzero there exists such that and .
- Each absolutely irreducible polynomial has infinitely many -rational points.
- If is a finitely generated integral domain over with quotient field which is regular over , then there exist a homomorphism such that for each
![{\displaystyle f\in K[T_{1},T_{2},\cdots ,T_{r},X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7cc9e7f828bb08fb8f7b13291ab4e7498463de)
![{\displaystyle g\in K[T_{1},T_{2},\cdots ,T_{r}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85f1b7e40db6235a8370929b513c516834bfd20c)
![{\displaystyle f\in K[T,X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125b03cfefb3238adea8445e586ea2540927dda5)
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 of a field is profinite, hence compact, and hence equipped with a normalized Haar measure. Let be a countable Hilbertian field and let be a positive integer. Then for almost all -tuple , 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
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