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| In [[mathematics]], a [[field (mathematics)|field]] ''F'' is called '''quasi-algebraically closed''' (or '''C<sub>1</sub>''') if every non-constant [[homogeneous polynomial]] ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by [[C. C. Tsen]], a student of [[Emmy Noether]] in a 1936 paper; and later in the 1951 [[Princeton University]] dissertation of [[Serge Lang]]. The idea itself is attributed to Lang's advisor [[Emil Artin]].
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| Formally, if ''P'' is a non-constant homogeneous polynomial in variables
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| :''X''<sub>1</sub>, ..., ''X''<sub>''N''</sub>,
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| and of degree ''d'' satisfying
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| :''d'' < ''N''
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| then it has a non-trivial zero over ''F''; that is, for some ''x''<sub>''i''</sub> in ''F'', not all 0, we have
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| :''P''(''x''<sub>''1''</sub>, ..., ''x''<sub>''N''</sub>) = 0.
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| In geometric language, the [[hypersurface]] defined by ''P'', in [[projective space]] of dimension ''N'' − 2, then has a point over ''F''.
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| ==Examples==
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| *Any [[algebraically closed field]] is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.<ref name=FJ455>Fried & Jarden (2008) p.455</ref>
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| *Any [[finite field]] is quasi-algebraically closed by the [[Chevalley–Warning theorem]].<ref name=FJ456>Fried & Jarden (2008) p.456</ref><ref name=S79162>Serre (1979) p.162</ref><ref name=GS142>Gille & Szamuley (2006) p.142</ref>
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| *[[Algebraic function field]]s over algebraically closed fields are quasi-algebraically closed by [[Tsen's theorem]].<ref name=S79162/><ref name=GS143>Gille & Szamuley (2006) p.143</ref>
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| *The maximal unramified extension of a complete field with a discrete valuation and a [[perfect field|perfect]] residue field is quasi-algebraically closed.<ref name=S79162/>
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| *A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.<ref name=S79162/><ref name=GS144>Gille & Szamuley (2006) p.144</ref>
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| * A [[pseudo algebraically closed field]] of [[Characteristic (algebra)|characteristic]] zero is quasi-algebraically closed.<ref name=FJ462>Fried & Jarden (2008) p.462</ref>
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| ==Properties==
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| *Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.
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| *The [[Brauer group]] of a finite extension of a quasi-algebraically closed field is trivial.<ref>Lorenz (2008) p.181</ref><ref name=S79161>Serre (1979) p.161</ref><ref name=GS141>Gille & Szamuely (2006) p.141</ref>
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| *A quasi-algebraically closed field has [[Cohomological dimension of a field|cohomological dimension]] at most 1.<ref name=GS141/>
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| ==''C''<sub>k</sub> fields==
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| Quasi-algebraically closed fields are also called ''C''<sub>1</sub>. A '''C<sub>''k''</sub> field''', more generally, is one for which any homogeneous polynomial of degree ''d'' in ''N'' variables has a non-trivial zero, provided
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| :''d''<sup>''k''</sup> < ''N'',
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| for ''k'' ≥ 1.<ref name=SGC87>Serre (1997) p.87</ref> The condition was first introduced and studied by Lang.<ref name=GS141/> If a field is C<sub>i</sub> then so is a finite extension.<ref name=L245>Lang (1997) p.245</ref><ref name=SGC87/> The C<sub>0</sub> fields are precisely the algebraically closed fields.<ref name=NSW361/><ref name=Lor116>Lorenz (2008) p.116</ref>
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| Lang and Nagata proved that if a field is ''C''<sub>''k''</sub>, then any extension of [[transcendence degree]] ''n'' is ''C''<sub>''k''+''n''</sub>.<ref name=Lor119>Lorenz (2008) p.119</ref><ref name=SGC88>Serre (1997) p.88</ref><ref name=FJ459>Fried & Jarden (2008) p.459</ref> The smallest ''k'' such that ''K'' is a ''C''<sub>k</sub> field (<math>\infty</math> if no such number exists), is called the '''[[diophantine dimension]]''' ''dd''(''K'') of ''K''.<ref name=NSW361>{{cite book | title=Cohomology of Number Fields | volume=323 | series=Grundlehren der Mathematischen Wissenschaften | first1=Jürgen | last1=Neukirch | first2=Alexander | last2=Schmidt | first3=Kay | last3=Wingberg | edition=2nd | publisher=[[Springer-Verlag]] | year=2008 | isbn=3-540-37888-X | page=361}}</ref>
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| ===''C''<sub>2</sub> fields===
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| Every finite field is C<sub>2</sub>.<ref name=FJ462/>
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| ====Properties====
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| Suppose that the field ''k'' is ''C''<sub>2</sub>.
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| * Any skew field ''D'' finite over ''k'' as centre has the property that the [[reduced norm]] ''D''<sup>∗</sup> → ''k''<sup>∗</sup> is surjective.<ref name=SGC88/>
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| * Every quadratic form in 5 or more variables over ''k'' is [[Isotropic quadratic form|isotropic]].<ref name=SGC88/>
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| ====Artin's conjecture====
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| Artin conjectured that [[p-adic field|''p''-adic field]]s were ''C''<sub>2</sub>, but
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| [[Guy Terjanian]] found [[Ax–Kochen theorem|''p''-adic counterexamples]] for all ''p''.<ref>{{cite journal | first=Guy | last=Terjanian | authorlink=Guy Terjanian | title=Un contre-example à une conjecture d'Artin | journal=C. R. Acad. Sci. Paris Sér. A-B | volume=262 | page=A612 | year=1966 | zbl=0133.29705 | language=French }}</ref><ref name=L247>Lang (1997) p.247</ref> The [[Ax–Kochen theorem]] applied methods from [[model theory]] to show that Artin's conjecture was true for '''Q'''<sub>''p''</sub> with ''p'' large enough (depending on ''d''). | |
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| ===Weakly C<sub>''k''</sub> fields===
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| A field ''K'' is '''weakly C<sub>''k'',''d''</sub>''' if for every homogeneous polynomial of degree ''d'' in ''N'' variables satisfying
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| :''d''<sup>''k''</sup> < ''N'' | |
| the [[Zariski topology|Zariski closed]] set ''V''(''f'') of '''P'''<sup>''n''</sub>(''K'') contains a [[subvariety]] which is Zariski closed over ''K''.
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| A field which is weakly C<sub>''k'',''d''</sub> for every ''d'' is '''weakly C<sub>''k''</sub>'''.<ref name=FJ456/>
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| ====Properties====
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| * A C<sub>''k''</sub> field is weakly C<sub>''k''</sub>.<ref name=FJ456/>
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| * A [[perfect field|perfect]] PAC weakly C<sub>''k''</sub> field is C<sub>''k''</sub>.<ref name=FJ456/>
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| * A field ''K'' is weakly C<sub>''k'',''d''</sub> if and only if every form satisfying the conditions has a point '''x''' defined over a field which is a [[primary extension]] of ''K''.<ref name=FJ457>Fried & Jarden (2008) p.457</ref>
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| * If a field is weakly C<sub>''k''</sub>, then any extension of transcendence degree ''n'' is weakly C<sub>''k''+''n''</sub>.<ref name=FJ459/>
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| * Any extension of an algebraically closed field is weakly C<sub>1</sub>.<ref name=FJ461/>
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| * Any field with procyclic absolute Galois group is weakly C<sub>1</sub>.<ref name=FJ461/>
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| * Any field of positive characteristic is weakly C<sub>2</sub>.<ref name=FJ461/>
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| * If the field of rational numbers is weakly C<sub>1</sub>, then every field is weakly C<sub>1</sub>.<ref name=FJ461>Fried & Jarden (2008) p.461</ref>
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| ==See also==
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| * [[Brauer's theorem on forms]]
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| * [[Tsen rank]]
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| ==References==
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| {{reflist|2}}
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| *{{cite journal | first1=James | last1=Ax | author1-link=James Ax | first2=Simon | last2=Kochen | author2-link=Simon B. Kochen | title=Diophantine problems over local fields I | title=Amer. J. Math. | volume=87 | pages=605–630 | year=1965 | zbl=0136.32805 }}
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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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| * {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
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| * {{cite book | zbl=0185.08304 | last=Greenberg | first=M.J. | title=Lectures of forms in many variables | series=Mathematics Lecture Note Series | location=New York-Amsterdam | publisher=W.A. Benjamin | year=1969 | zbl=0185.08304| series=Mathematics Lecture Note Series }}
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| * {{cite journal | zbl=0046.26202 | last=Lang | first=Serge | authorlink=Serge Lang | title=On quasi algebraic closure | journal=[[Annals of Mathematics]] | volume=55 | year=1952 | pages=373–390 }}
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| * {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine Geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
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| * {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | pages=109-126 | zbl=1130.12001 }}
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| * {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local fields | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 }}
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| * {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Galois cohomology | publisher=[[Springer-Verlag]] | year=1997| isbn=3-540-61990-9 | zbl=0902.12004 }}
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| *{{cite journal | first=C. | last=Tsen | authorlink=C. C. Tsen | title=Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper | journal=J. Chinese Math. Soc. | volume=171 | year=1936 | pages=81–92 | zbl=0015.38803 }}
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| [[Category:Field theory]]
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| [[Category:Diophantine geometry]]
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I am Oscar and I totally dig that title. Minnesota is where he's been residing for many years. The favorite pastime for my kids and me is to perform baseball and I'm trying to make it a occupation. He used to be unemployed but now he is a pc operator but his marketing by no means arrives.
Review my web-site: http://ironptstudio.com/rev/239375