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| 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>
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| ==Formulation==
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| A field ''K'' is pseudo algebraically closed (usually abbreviated by '''PAC'''<ref name=MF192/>) if one of the following equivalent conditions holds:
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| *Each [[absolutely irreducible]] variety <math>V</math> defined over <math>K</math> has a <math>K</math>-[[rational point]].
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| *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>.
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| *Each absolutely irreducible polynomial <math>f\in K[T,X]</math> has infinitely many <math>K</math>-rational points.
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| *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>
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| == Examples ==
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| * Algebraically closed fields and [[separably closed]] fields are always PAC.
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| *[[Pseudo-finite field]]s and [[hyper-finite field]]s are PAC.
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| * 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/>
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| * Infinite [[algebraic extension]]s of [[finite field]]s are PAC.<ref name=MF196>Fried & Jarden (2008) p.196</ref>
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| * ''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.)
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| * 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.
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| ==Properties==
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| * 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/>
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| * 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>
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| * A PAC field of [[Characteristic (algebra)|characteristic]] zero is [[C1 field|C1]].<ref name=MF462>Fried & Jarden (2008) p.462</ref>
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| == References ==
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| {{reflist}}
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| * {{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 }}
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| [[Category:Algebraic geometry]]
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| [[Category:Field theory]]
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