Quasi-algebraically closed field
In mathematics, a field F is called quasi-algebraically closed 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 xi in F, not all 0, we have
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.
''C''''k'' 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, providedfor 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, is called the diophantine dimension dd of K.
''C''1 fields
Every finite field is C1.''C''2 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, butGuy 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.
Weakly ''C''''k'' fields
A field K is weakly Ck,''d if for every homogeneous polynomial of degree d'' in N variables satisfyingthe Zariski closed set V of Pn contains a subvariety which is Zariski closed over K.
A field that is weakly Ck,''d for every d'' is weakly Ck.
Properties
- A Ck field is weakly Ck.
- A perfect PAC weakly Ck field is Ck.
- A field K is weakly Ck,''d if and only if every form satisfying the conditions has a point x defined over a field which is a primary extension of K''.
- If a field is weakly Ck, then any extension of transcendence degree n is weakly Ck+''n.
- Any extension of an algebraically closed field is weakly C''1.
- Any field with procyclic absolute Galois group is weakly C1.
- Any field of positive characteristic is weakly C2.
- If the field of rational numbers and the function fields are weakly C1, then every field is weakly C1.