Quasi-algebraically Closed Field - Ck Fields

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 ≥ 1. 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 ( if no such number exists), is called the diophantine dimension dd(K) of K.

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