Quasi-algebraically closed field explained

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

X₁, ..., X_N,and of degree d satisfying

d < Nthen 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, 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.^[1]
• Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.^{[2] [3] [4]}
• Algebraic function fields of dimension 1 over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.^{[3] [5]}
• The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.^[3]
• A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.^{[3] [6]}
• A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.^[7]

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.^{[8] [9] [10]}
• A quasi-algebraically closed field has cohomological dimension at most 1.^[10]

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.^[11] The condition was first introduced and studied by Lang.^[10] If a field is C_i then so is a finite extension.^{[11] [12]} The C₀ fields are precisely the algebraically closed fields.^[13]

Lang and Nagata proved that if a field is C_k, then any extension of transcendence degree n is C_k+n.^{[14] [15] [16]} The smallest k such that K is a C_k field (

inf ty if no such number exists), is called the diophantine dimension dd(K) of K.^[17]

C₁ fields

Every finite field is C₁.^[7]

C₂ 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^&lowast; → k^&lowast; is surjective.^[15]
• Every quadratic form in 5 or more variables over k is isotropic.^[15]

Artin's conjecture

Artin conjectured that p-adic fields were C₂, but Guy Terjanian found p-adic counterexamples for all p.^{[18] [19]} 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 < Nthe 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.^[2]

Properties

• A C_k field is weakly C_k.^[2]
• A perfect PAC weakly C_k field is C_k.^[2]
• 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.^[20]
• If a field is weakly C_k, then any extension of transcendence degree n is weakly C_k+n.^[16]
• 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

Q and the function fields F _{p(t)} are weakly C₁, then every field is weakly C₁.^[21]

See also

• Brauer's theorem on forms
• Tsen rank

References

• James . Ax . James Ax . Simon . Kochen . Simon B. Kochen . Diophantine problems over local fields I . Amer. J. Math. . 87 . 605–630 . 1965 . 3 . 0136.32805 . 10.2307/2373065. 2373065 .
• Book: Fried . Michael D. . Jarden . Moshe . Field arithmetic . 3rd revised . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge . 11 . . 2008 . 978-3-540-77269-9 . 1145.12001 .
• Book: Gille . Philippe . Szamuely . Tamás . Central simple algebras and Galois cohomology . Cambridge Studies in Advanced Mathematics . 101 . Cambridge . . 2006 . 0-521-86103-9 . 1137.12001 .
• Book: Greenberg, M.J. . Lectures of forms in many variables . New York-Amsterdam . W.A. Benjamin . 1969 . 0185.08304. Mathematics Lecture Note Series .
  • Book: Lang, Serge . Serge Lang

. Serge Lang . Survey of Diophantine Geometry . . 1997 . 3-540-61223-8 . 0869.11051 .

• Book: Lorenz, Falko . Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics . 2008 . Springer . 978-0-387-72487-4 . 109–126 . 1130.12001 .
• Book: Serre, Jean-Pierre . Jean-Pierre Serre

. Jean-Pierre Serre . . Marvin Jay . Greenberg . Marvin Greenberg. . 67 . . 1979 . 0-387-90424-7 . 0423.12016 .

• Book: Serre, Jean-Pierre . Jean-Pierre Serre

. Jean-Pierre Serre . Galois cohomology . . 1997. 3-540-61990-9 . 0902.12004 .

Notes and References

1. Fried & Jarden (2008) p. 455
2. Fried & Jarden (2008) p. 456
3. Serre (1979) p. 162
4. Gille & Szamuley (2006) p. 142
5. Gille & Szamuley (2006) p. 143
6. Gille & Szamuley (2006) p. 144
7. Fried & Jarden (2008) p. 462
8. Lorenz (2008) p. 181
9. Serre (1979) p. 161
10. Gille & Szamuely (2006) p. 141
11. Serre (1997) p. 87
12. Lang (1997) p. 245
13. Lorenz (2008) p. 116
14. Lorenz (2008) p. 119
15. Serre (1997) p. 88
16. Fried & Jarden (2008) p. 459
17. Book: Cohomology of Number Fields . 323 . Grundlehren der Mathematischen Wissenschaften . Jürgen . Neukirch . Alexander . Schmidt . Kay . Wingberg . 2nd . . 2008 . 978-3-540-37888-4 . 361.
18. Guy . Terjanian . Guy Terjanian . Un contre-example à une conjecture d'Artin . Comptes Rendus de l'Académie des Sciences, Série A-B . 262 . A612 . 1966 . 0133.29705 . French .
19. Lang (1997) p. 247
20. Fried & Jarden (2008) p. 457
21. Fried & Jarden (2008) p. 461