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.
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. 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.
[13][14] 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.[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).