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Concept
Quasi-algebraically closed field
Summary
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 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.
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.
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.
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