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 and of degree d satisfying then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have 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 for k ≥ 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.

C2 fields

Properties

Suppose that the field k is C2.

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 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

Citations