Квазижабық емес өрістер және олардың қасиеттері

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
X1, , XN,
and of degree d satisfying
d < N
then 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.

Any algebraically closed field is quasi algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non trivial zero. Any finite field is quasi algebraically closed by the Chevalley–Warning theorem. Algebraic function fields of dimension 1 over algebraically closed fields are quasi algebraically closed by Tsen's theorem. The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi algebraically closed. A pseudo algebraically closed field of characteristic zero is quasi algebraically closed.

Any algebraic extension of a quasi algebraically closed field is quasi algebraically closed. The Brauer group of a finite extension of a quasi algebraically closed field is trivial. A quasi algebraically closed field has cohomological dimension at most 1. The condition was first introduced and studied by Lang. 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.

Every finite field is C1. 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).