Quasi-algebraically closed field#C2 fields

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 {{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

: X₁, ..., 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(x₁, ..., x_N) = 0.

In geometric language, the hypersurface defined by P, in projective space of degree {{nowrap|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.Lorenz (2008) p. 181Serre (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 C₁. 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.{{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}}

= ''C''1 fields =

Every finite field is C₁.

= ''C''2 fields =

== Properties ==

Suppose that the field k is C₂.

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 C₂, but

Guy Terjanian found p-adic counterexamples for all p.{{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 }}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 Pⁿ(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.

== 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 $\mathbb{Q}$ and the function fields $\mathbb{F}_p(t)$ are weakly C₁, then every field is weakly C₁.Fried & Jarden (2008) p. 461

Citations

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 | 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 (mathematics)

Category:Diophantine geometry

Quasi-algebraically closed field#C2 fields

Examples

Properties

''C''<sub>''k''</sub> fields

= ''C''<sub>1</sub> fields =

= ''C''<sub>2</sub> fields =

== Properties ==

== Artin's conjecture ==

= Weakly ''C''<sub>''k''</sub> fields =

== Properties ==

See also

Citations

References