Quasi-algebraically closed field

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In mathematics, a field F is called quasi-algebraically closed (or C1) if for every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree.

In other words, if P is a non-constant homogeneous polynomial in indeterminates

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 dimension N − 1, then has a point over F.

[edit] Examples and properties

  • Any finite field is quasi-algebraically closed. (Chevalley-Warning theorem)
  • Function fields of algebraic curves over algebraically closed fields are quasi-algebraically closed.(Tsen's theorem).
  • Lang showed that if K is a complete field with a discrete valuation and an algebraically closed residue field, then K is quasi-algebraically closed.
  • Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.

The Brauer group of a quasi-algebraically closed field is trivial.

The idea of quasi-algebraically closed fields was investigated by C. Tsen, a student of Emmy Noether in a 1936 paper; and later in the 1951 Princeton University dissertation of Serge Lang. The idea itself is attributed to Lang's advisor Emil Artin.

[edit] 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.

Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree n is Ck+n.

Artin conjectured that p-adic fields were C2, but Guy Terjanian found p-adic counterexamples for all p. 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).

[edit] References

  • C. Tsen, Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer K"orper, J. Chinese Math. Soc. 171 (1936), 81-92
  • Serge Lang, On quasi algebraic closure, Annals of Mathematics 55 (1952), 373–390.
  • M. J. Greenberg , Lectures on Forms in Many Variables, Benjamin, 1969.
  • J. Ax and S. Kochen. Diophantine problems over local fields I Amer. J. Math., 87:605-630, 1965
  • J.-P. Serre, Galois cohomology, ISBN 3-540-61990-9
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