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.

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

X₁, ..., X_N,

and of degree d satisfying

d < N

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

Examples and properties

• 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.
• Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.
• Function fields of algebraic curves over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.
• If K is a complete field with a discrete valuation and an algebraically closed residue field, then K is quasi-algebraically closed by a result of Lang.
• 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. 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.

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.

Lang and Nagata proved that if a field is C_k, then any extension of transcendence degree n is C_k+n. The smallest k such that K is a C_k field ($\infty$ if no such number exists), is called the diophantine dimension dd(K) of K.

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 Q_p with p large enough (depending on d).

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