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Quasi-algebraically closed field

In mathematics, a field F is called quasi-algebraically closed (or C<sub>1</sub>) 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

X<sub>1</sub>, ..., X<sub>N</sub>,

and of degree d satisfying

d < N

then it has a non-trivial zero over F; that is, for some x<sub>i</sub> in F, not all 0, we have

P(x<sub>1</sub>, ..., x<sub>N</sub>) = 0.

In geometric language, the hypersurface defined by P, in projective space of degree , then has a point over F.

Examples

Properties

C<sub>k</sub> fields

Quasi-algebraically closed fields are also called C<sub>1</sub>. A C<sub>k</sub> field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided

d<sup>k</sup> < N,

for k &ge; 1. The condition was first introduced and studied by Lang. If a field is C<sub>i</sub> then so is a finite extension. The C<sub>0</sub> fields are precisely the algebraically closed fields.

Lang and Nagata proved that if a field is C<sub>k</sub>, then any extension of transcendence degree n is C<sub>k+n</sub>. The smallest k such that K is a C<sub>k</sub> field ( if no such number exists), is called the diophantine dimension dd(K) of K.

C<sub>1</sub> fields

Every finite field is C<sub>1</sub>.

C<sub>2</sub> fields

Properties

Suppose that the field k is C<sub>2</sub>.

  • Any skew field D finite over k as centre has the property that the reduced norm D<sup>&lowast;</sup> → k<sup>&lowast;</sup> is surjective.
  • Every quadratic form in 5 or more variables over k is isotropic.

Artin's conjecture

Artin conjectured that p-adic fields were C<sub>2</sub>, 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 Q<sub>p</sub> with p large enough (depending on d).

Weakly C<sub>k</sub> fields

A field K is weakly C<sub>k,d</sub> if for every homogeneous polynomial of degree d in N variables satisfying

d<sup>k</sup> < N

the Zariski closed set V(f) of P<sup>n</sup>(K) contains a subvariety which is Zariski closed over K.

A field that is weakly C<sub>k,d</sub> for every d is weakly C<sub>k</sub>.

Properties

  • A C<sub>k</sub> field is weakly C<sub>k</sub>.
  • A perfect PAC weakly C<sub>k</sub> field is C<sub>k</sub>.
  • A field K is weakly C<sub>k,d</sub> 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 C<sub>k</sub>, then any extension of transcendence degree n is weakly C<sub>k+n</sub>.
  • Any extension of an algebraically closed field is weakly C<sub>1</sub>.
  • Any field with procyclic absolute Galois group is weakly C<sub>1</sub>.
  • Any field of positive characteristic is weakly C<sub>2</sub>.
  • If the field of rational numbers and the function fields are weakly C<sub>1</sub>, then every field is weakly C<sub>1</sub>.

See also

Citations

References