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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

  • 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]

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 which 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

  • Brauer's theorem on forms
  • Tsen rank

Citations

  1. ^ Fried & Jarden (2008) p.455
  2. ^ a b c d Fried & Jarden (2008) p.456
  3. ^ a b c d Serre (1979) p.162
  4. ^ Gille & Szamuley (2006) p.142
  5. ^ Gille & Szamuley (2006) p.143
  6. ^ Gille & Szamuley (2006) p.144
  7. ^ a b Fried & Jarden (2008) p.462
  8. ^ Lorenz (2008) p.181
  9. ^ Serre (1979) p.161
  10. ^ a b c Gille & Szamuely (2006) p.141
  11. ^ a b Serre (1997) p.87
  12. ^ Lang (1997) p.245
  13. ^ a b Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X.
  14. ^ Lorenz (2008) p.116
  15. ^ Lorenz (2008) p.119
  16. ^ a b c Serre (1997) p.88
  17. ^ a b 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. ^ a b c d Fried & Jarden (2008) p.461

References

  • Ax, James; Kochen, Simon (1965). "Diophantine problems over local fields I". Amer. J. Math. 87: 605–630. doi:10.2307/2373065. Zbl 0136.32805.
  • Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
  • Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
  • Greenberg, M.J. (1969). Lectures of forms in many variables. Mathematics Lecture Note Series. New York-Amsterdam: W.A. Benjamin. Zbl 0185.08304.
  • Lang, Serge (1952), "On quasi algebraic closure", Annals of Mathematics, 55: 373–390, doi:10.2307/1969785, Zbl 0046.26202
  • Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
  • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 109–126. ISBN 978-0-387-72487-4. Zbl 1130.12001.
  • Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016.
  • Serre, Jean-Pierre (1997). Galois cohomology. Springer-Verlag. ISBN 3-540-61990-9. Zbl 0902.12004.
  • Tsen, C. (1936), "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper", J. Chinese Math. Soc., 171: 81–92, Zbl 0015.38803
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