# Quasi-algebraically closed field

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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 Template:Harv; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper Template:Harv. 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 dimension N − 2, then has a point over F.

## 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 ($\infty$ if no such number exists), is called the diophantine dimension dd(K) of K.

### C2 fields

Every finite field is C2.

#### Properties

Suppose that the field k is C2.

• Any skew field D finite over k as centre has the property that the reduced norm Dk 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 weakly Ck,d if for every homogeneous polynomial of degree d in N variables satisfying

dk < N

the Zariski closed set V(f) of Pn(K) contains a subvariety which is Zariski closed over K.

A field which 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 C1.
• Any field with procyclic absolute Galois group is weakly C1.
• Any field of positive characteristic is weakly C2.
• If the field of rational numbers is weakly C1, then every field is weakly C1.