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.

Read more about this topic:  Quasi-algebraically Closed Field

Famous quotes containing the word fields:

    If the sight of the blue skies fills you with joy, if a blade of grass springing up in the fields has power to move you, if the simple things of nature have a message that you understand, rejoice, for your soul is alive ...
    Eleonora Duse (1859–1924)