Quasi-algebraically Closed Field - C_k Fields

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

dk < N,

for k ≥ 1. If a field is C_i then so is a finite extension. The C₀ fields are precisely the algebraically closed fields.

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 ( if no such number exists), is called the diophantine dimension dd(K) of K.