Quasi-algebraically closed field

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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.

Contents

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

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]

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∗ → 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. 1 2 3 4 Fried & Jarden (2008) p. 456
  3. 1 2 3 4 Serre (1979) p. 162
  4. ↑ Gille & Szamuley (2006) p. 142
  5. ↑ Gille & Szamuley (2006) p. 143
  6. ↑ Gille & Szamuley (2006) p. 144
  7. 1 2 Fried & Jarden (2008) p. 462
  8. ↑ Lorenz (2008) p. 181
  9. ↑ Serre (1979) p. 161
  10. 1 2 3 Gille & Szamuely (2006) p. 141
  11. 1 2 Serre (1997) p. 87
  12. ↑ Lang (1997) p. 245
  13. 1 2 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. 1 2 3 Serre (1997) p. 88
  17. 1 2 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. 1 2 3 4 Fried & Jarden (2008) p. 461

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