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 (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (Lang 1952). The idea itself is attributed to Lang's advisor Emil Artin.
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Formally, if P is a non-constant homogeneous polynomial in variables
X1, ..., XN,and of degree d satisfying
d < Nthen it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have
P(x1, ..., xN) = 0.In geometric language, the hypersurface defined by P, in projective space of degree N − 2, then has a point over F.
Examples
Properties
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. 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 Ck, then any extension of transcendence degree n is Ck+n. The smallest k such that K is a Ck field (
C1 fields
Every finite field is C1.
Properties
Suppose that the field k is C2.
Artin's conjecture
Artin conjectured that p-adic fields were C2, 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 Qp with p large enough (depending on d).
Weakly Ck fields
A field K is weakly Ck,d if for every homogeneous polynomial of degree d in N variables satisfying
dk < Nthe Zariski closed set V(f) of Pn(K) contains a subvariety which is Zariski closed over K.
A field which is weakly Ck,d for every d is weakly Ck.