Quasi-algebraically closed field

In mathematics, a field ''F'' is called '''quasi-algebraically closed''' (or '''''C''₁''') 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 {{harv|Tsen|1936}}; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper {{harv|Lang|1952}}. The idea itself is attributed to Lang's advisor Emil Artin.

Formally, if ''P'' is a non-constant homogeneous polynomial in variables : ''X''₁, ..., ''X''_''N'', and of degree ''d'' satisfying : ''d'' < ''N'' then it has a non-trivial zero over ''F''; that is, for some ''x''_''i'' in ''F'', not all 0, we have : ''P''(''x''₁, ..., ''x''_''N'') = 0.

In geometric language, the hypersurface defined by ''P'', in projective space of degree {{nowrap|''N'' &minus; 2}}, then has a point over ''F''.

== Examples == * 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> * 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> * Algebraic function fields of dimension 1 over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.<ref name=S79162/><ref name=GS143>Gille & Szamuley (2006) p. 143</ref> * The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.<ref name=S79162/> * 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> * A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.<ref name=FJ462>Fried & Jarden (2008) p. 462</ref>

== Properties == * Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed. * 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> * A quasi-algebraically closed field has cohomological dimension at most 1.<ref name=GS141/>

== ''C''_''k'' fields == Quasi-algebraically closed fields are also called ''C''₁. A '''''C''_''k'' field''', more generally, is one for which any homogeneous polynomial of degree ''d'' in ''N'' variables has a non-trivial zero, provided : ''d''^''k'' < ''N'', for ''k'' &ge; 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''_''i'' then so is a finite extension.<ref name=SGC87/><ref name=L245>Lang (1997) p. 245</ref> The ''C''₀ fields are precisely the algebraically closed fields.<ref name=NSW361/><ref name=Lor116>Lorenz (2008) p. 116</ref>

Lang and Nagata proved that if a field is ''C''_''k'', then any extension of transcendence degree ''n'' is ''C''_''k''+''n''.<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''_''k'' 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=978-3-540-37888-4 | page=361}}</ref>

=== ''C''₁ fields ===

Every finite field is ''C''₁.<ref name=FJ462/>

=== ''C''₂ fields ===

==== Properties ==== Suppose that the field ''k'' is ''C''₂. * Any skew field ''D'' finite over ''k'' as centre has the property that the reduced norm ''D''^&lowast; → ''k''^&lowast; is surjective.<ref name=SGC88/> * Every quadratic form in 5 or more variables over ''k'' is isotropic.<ref name=SGC88/>

==== Artin's conjecture ==== Artin conjectured that ''p''-adic fields were ''C''₂, but Guy Terjanian found ''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=Comptes Rendus de l'Académie des Sciences, Série 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'''_''p'' with ''p'' large enough (depending on ''d'').

=== Weakly ''C''_''k'' fields === A field ''K'' is '''weakly ''C''_''k'',''d''''' if for every homogeneous polynomial of degree ''d'' in ''N'' variables satisfying : ''d''^''k'' < ''N'' the Zariski closed set ''V''(''f'') of '''P'''^''n''(''K'') contains a subvariety which is Zariski closed over ''K''.

A field that is weakly ''C''_''k'',''d'' for every ''d'' is '''weakly ''C''_''k'''''.<ref name=FJ456/>

==== Properties ==== * A ''C''_''k'' field is weakly ''C''_''k''.<ref name=FJ456/> * A perfect PAC weakly ''C''_''k'' field is ''C''_''k''.<ref name=FJ456/> * A field ''K'' is weakly ''C''_''k'',''d'' 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> * If a field is weakly ''C''_''k'', then any extension of transcendence degree ''n'' is weakly ''C''_''k''+''n''.<ref name=FJ459/> * Any extension of an algebraically closed field is weakly ''C''₁.<ref name=FJ461/> * Any field with procyclic absolute Galois group is weakly ''C''₁.<ref name=FJ461/> * Any field of positive characteristic is weakly ''C''₂.<ref name=FJ461/> * If the field of rational numbers <math>\mathbb{Q}</math> and the function fields <math>\mathbb{F}_p(t)</math> are weakly ''C''₁, then every field is weakly ''C''₁.<ref name=FJ461>Fried & Jarden (2008) p. 461</ref>

== See also == * Brauer's theorem on forms * Tsen rank

== Citations == {{reflist|2}}

== References == * {{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 | journal=Amer. J. Math. | volume=87 | pages=605–630 | year=1965 | issue=3 | zbl=0136.32805 | doi=10.2307/2373065| jstor=2373065 }} * {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe |author-link=Michael D. Fried |author-link2=Moshe Jarden | 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 }} * {{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 }} * {{cite book | last=Greenberg | first=M.J. | title=Lectures of forms in many variables | location=New York-Amsterdam | publisher=W.A. Benjamin | year=1969 | zbl=0185.08304| series=Mathematics Lecture Note Series }} * {{citation | zbl=0046.26202 | last=Lang | first=Serge | authorlink=Serge Lang | title=On quasi algebraic closure | journal=Annals of Mathematics | volume=55 | year=1952 | issue=2 | pages=373–390 | doi=10.2307/1969785| jstor=1969785 }} * {{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 }} * {{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 }} * {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local Fields | translator-first1= Marvin Jay |translator-last1=Greenberg |translator-link=Marvin Greenberg| series=Graduate Texts in Mathematics | volume=67 | publisher=Springer-Verlag | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 }} * {{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 }} * {{citation | 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 }}

Category:Field theory Category:Diophantine geometry