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.
Read more about this topic: Quasi-algebraically Closed Field
Famous quotes containing the word fields:
“Like a man traveling in foggy weather, those at some distance before him on the road he sees wrapped up in the fog, as well as those behind him, and also the people in the fields on each side, but near him all appears clear, though in truth he is as much in the fog as any of them.”
—Benjamin Franklin (1706–1790)