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 ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper. 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 &minus; 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.
 * Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.
 * Algebraic function fields of dimension 1 over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.
 * The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.
 * A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.
 * A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.

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.
 * A quasi-algebraically closed field has cohomological dimension at most 1.

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 &ge; 1. The condition was first introduced and studied by Lang. If a field is Ci then so is a finite extension. The C0 fields are precisely the algebraically closed fields.

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

C1 fields
Every finite field is C1.

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&lowast; → k&lowast; is surjective.
 * Every quadratic form in 5 or more variables over k is isotropic.

Artin's conjecture
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).

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.

Properties

 * A Ck field is weakly Ck.
 * A perfect PAC weakly Ck field is Ck.
 * 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.
 * If a field is weakly Ck, then any extension of transcendence degree n is weakly Ck+n.
 * Any extension of an algebraically closed field is weakly C1.
 * Any field with procyclic absolute Galois group is weakly C1.
 * Any field of positive characteristic is weakly C2.
 * If the field of rational numbers $$\mathbb{Q}$$ and the function fields $$\mathbb{F}_p(t)$$ are weakly C1, then every field is weakly C1.