Quasi-algebraically closed field

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. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . The idea itself is attributed to Lang's advisor Emil Artin.
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 "x""i" 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".
"C"k fields.
Quasi-algebraically closed fields are also called "C"1. 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
for "k" ≥ 1. The condition was first introduced and studied by Lang. 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 "C""k", then any extension of transcendence degree "n" is "C""k"+"n". The smallest "k" such that "K" is a "C"k field (formula_1 if no such number exists), is called the diophantine dimension "dd"("K") of "K".
"C"1 fields.
Every finite field is C1.
"C"2 fields.
Properties.
Suppose that the field "k" is "C"2.
Artin's conjecture.
Artin conjectured that "p"-adic fields were "C"2, but 
Guy Terjanian found "p"-adic counterexamples for all "p". The Ax–Kochen theorem applied methods from model theory to show that Artin's conjecture was true for Q"p" with "p" large enough (depending on "d").
Weakly C"k" fields.
A field "K" is weakly C"k","d" if for every homogeneous polynomial of degree "d" in "N" variables satisfying
the Zariski closed set "V"("f") of P"n"("K") contains a subvariety which is Zariski closed over "K".
A field which is weakly C"k","d" for every "d" is weakly C"k".