# Quasi-algebraically closed field

In mathematics, a field F is called quasi-algebraically closed (or C$_1$) 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 (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (Lang 1952). The idea itself is attributed to Lang's advisor Emil Artin.
Formally, if P is a non-constant homogeneous polynomial in variables
    X$_1$, ..., X$_N$,
and of degree d satisfying
    d < N
then it has a non-trivial zero over F; that is, for some x$_i$ in F, not all 0, we have
    P(x$_1$, ..., x$_N$) = 0.
In geometric language, the hypersurface defined by P, in projective space of degree N − 2, 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 C$_1$. A C$_k$ field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided
    d$^k$ < N,
for k ≥ 1. The condition was first introduced and studied by Lang. If a field is C$_i$ then so is a finite extension. The C$_0$ fields are precisely the algebraically closed fields.
Lang and Nagata proved that if a field is C$_k$, then any extension of transcendence degree n is C$_{k+n}$. The smallest k such that K is a C$_k$ field ($\infty$ if no such number exists), is called the diophantine dimension dd(K) of K.

### C$_1$ fields

Every finite field is C$_1$.

### C$_2$ fields
#### Properties

Suppose that the field k is C$_2$.
* 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 C$_2$, 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$_p$ with p large enough (depending on d).

### Weakly C$_k$ fields

A field K is weakly C$_{k,d}$ if for every homogeneous polynomial of degree d in N variables satisfying
    d$^k$ < N
the Zariski closed set V(f) of P$^n$(K) contains a subvariety which is Zariski closed over K.
A field that is weakly C$_{k,d}$ for every d is weakly C$_k$.

#### Properties

* A C$_k$ field is weakly C$_k$.
* A perfect PAC weakly C$_k$ field is C$_k$.
* A field K is weakly C$_{k,d}$ 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 C$_k$, then any extension of transcendence degree n is weakly C$_{k+n}$.
* Any extension of an algebraically closed field is weakly C$_1$.
* Any field with procyclic absolute Galois group is weakly C$_1$.
* Any field of positive characteristic is weakly C$_2$.
* If the field of rational numbers $\mathbb {Q}$ and the function fields $\mathbb {F} _{p}(t)$ are weakly C$_1$, then every field is weakly C$_1$.