quasi-algebraically closed field

In mathematics, a field ''F'' is called quasi-algebraically closed (or C₁) 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 by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . 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 ''N'' − 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.Fried & Jarden (2008) p.455
*Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.Fried & Jarden (2008) p.456Serre (1979) p.162Gille & Szamuley (2006) p.142
*Algebraic function fields of dimension 1 over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.Gille & Szamuley (2006) p.143
*The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.
*A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.Gille & Szamuley (2006) p.144
* A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.Fried & Jarden (2008) p.462

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.Serre (1979) p.161Gille & Szamuely (2006) p.141
*A quasi-algebraically closed field has cohomological dimension at most 1.

''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'' ≥ 1.Serre (1997) p.87 The condition was first introduced and studied by Lang. If a field is C_i then so is a finite extension.Lang (1997) p.245 The C₀ fields are precisely the algebraically closed fields.Lorenz (2008) p.116

Lang and Nagata proved that if a field is ''C''_''k'', then any extension of transcendence degree ''n'' is ''C''_''k''+''n''.Lorenz (2008) p.119Serre (1997) p.88Fried & Jarden (2008) p.459 The smallest ''k'' such that ''K'' is a ''C''_k field ($\infty$ if no such number exists), is called the diophantine dimension ''dd''(''K'') of ''K''.

''C''₁ fields

Every finite field is C₁.

''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''^∗ → ''k''^∗ is surjective.
* Every quadratic form in 5 or more variables over ''k'' is isotropic.

Artin's conjecture

Artin conjectured that ''p''-adic fields were ''C''₂, but
Guy Terjanian found ''p''-adic counterexamples for all ''p''.Lang (1997) p.247 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 which is weakly C_''k'',''d'' for every ''d'' is weakly C_''k''.

Properties

* A C_''k'' field is weakly C_''k''.
* A perfect PAC weakly C_''k'' field is C_''k''.
* 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''.Fried & Jarden (2008) p.457
* If a field is weakly C_''k'', then any extension of transcendence degree ''n'' is weakly C_''k''+''n''.
* Any extension of an algebraically closed field is weakly C₁.
* Any field with procyclic absolute Galois group is weakly C₁.
* Any field of positive characteristic is weakly C₂.
* If the field of rational numbers $\mathbb$ and the function fields $\mathbb_p(t)$ are weakly C₁, then every field is weakly C₁.Fried & Jarden (2008) p.461

See also

* Brauer's theorem on forms
* Tsen rank

Citations

References

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*{{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

Field (mathematics)
Diophantine geometry