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 (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (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, ..., XN,

and of degree d satisfying

d < N

then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have

P(x1, ..., xN) = 0.

In geometric language, the hypersurface defined by P, in projective space of degree N − 2, then has a point over F.

Examples

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]

Ck fields

Quasi-algebraically closed fields are also called C1. A Ck field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided

dk < N,

for k ≥ 1.[11] The condition was first introduced and studied by Lang.[10] If a field is Ci then so is a finite extension.[11][12] The C0 fields are precisely the algebraically closed fields.[13][14]

Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree n is Ck+n.[15][16][17] The smallest k such that K is a Ck field ( if no such number exists), is called the diophantine dimension dd(K) of K.[13]

C1 fields

Every finite field is C1.[7]

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.[16]
  • Every quadratic form in 5 or more variables over k is isotropic.[16]

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 Qp with p large enough (depending on d).

Weakly Ck fields

A field K is weakly Ck,d if for every homogeneous polynomial of degree d in N variables satisfying

dk < N

the Zariski closed set V(f) of Pn(K) contains a subvariety which is Zariski closed over K.

A field that is weakly Ck,d for every d is weakly Ck.[2]

Properties
  • A Ck field is weakly Ck.[2]
  • A perfect PAC weakly Ck field is Ck.[2]
  • A field K is weakly Ck,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 Ck, then any extension of transcendence degree n is weakly Ck+n.[17]
  • Any extension of an algebraically closed field is weakly C1.[21]
  • Any field with procyclic absolute Galois group is weakly C1.[21]
  • Any field of positive characteristic is weakly C2.[21]
  • If the field of rational numbers and the function fields are weakly C1, then every field is weakly C1.[21]

See also

Citations
  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. Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 978-3-540-37888-4.
  14. Lorenz (2008) p. 116
  15. Lorenz (2008) p. 119
  16. Serre (1997) p. 88
  17. Fried & Jarden (2008) p. 459
  18. Terjanian, Guy (1966). "Un contre-example à une conjecture d'Artin". Comptes Rendus de l'Académie des Sciences, Série A-B (in French). 262: A612. Zbl 0133.29705.
  19. Lang (1997) p. 247
  20. Fried & Jarden (2008) p. 457
  21. Fried & Jarden (2008) p. 461

References
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