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 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 , then has a point over F. Examples 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 complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang. A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed. Properties 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. 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 ( if no such number exists), is called the diophantine dimension dd(K) of K. C1 fields Every finite field is C1. C2 fields Properties Suppose that the field k is C2. Any skew field D finite over k as centre has the property that the reduced norm D∗ → k∗ is surjective. Every quadratic form in 5 or more variables over k is isotropic. 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