# 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 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

*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 dimension *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.
^{[1]} - Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.
^{[2]}^{[3]}^{[4]} - Algebraic function fields over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.
^{[3]}^{[5]} - The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.
^{[3]} - A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.
^{[3]}^{[6]} - A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.
^{[7]}

## 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.
^{[8]}^{[9]}^{[10]} - A quasi-algebraically closed field has cohomological dimension at most 1.
^{[10]}

*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.^{[11]} The condition was first introduced and studied by Lang.^{[10]} If a field is C_{i} then so is a finite extension.^{[11]}^{[12]} The C_{0} fields are precisely the algebraically closed fields.^{[13]}^{[14]}

Lang and Nagata proved that if a field is *C*_{k}, then any extension of transcendence degree *n* is *C*_{k+n}.^{[15]}^{[16]}^{[17]} The smallest *k* such that *K* is a *C*_{k} field ( if no such number exists), is called the **diophantine dimension** *dd*(*K*) of *K*.^{[13]}

*C*_{2} fields

Every finite field is C_{2}.^{[7]}

#### 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.^{[16]} - Every quadratic form in 5 or more variables over
*k*is isotropic.^{[16]}

#### Artin's conjecture

Artin conjectured that *p*-adic fields were *C*_{2}, but
Guy Terjanian found *p*-adic counterexamples for all *p*.^{[18]}^{[19]} 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 which is weakly C_{k,d} for every *d* is **weakly C _{k}**.

^{[2]}

#### Properties

- A C
_{k}field is weakly C_{k}.^{[2]} - A perfect PAC weakly C
_{k}field is C_{k}.^{[2]} - 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*.^{[20]} - If a field is weakly C
_{k}, then any extension of transcendence degree*n*is weakly C_{k+n}.^{[17]}

- Any extension of an algebraically closed field is weakly C
_{1}.^{[21]} - Any field with procyclic absolute Galois group is weakly C
_{1}.^{[21]} - Any field of positive characteristic is weakly C
_{2}.^{[21]}

- If the field of rational numbers is weakly C
_{1}, then every field is weakly C_{1}.^{[21]}

## See also

## Citations

- ↑ Fried & Jarden (2008) p.455
- ↑
^{2.0}^{2.1}^{2.2}^{2.3}Fried & Jarden (2008) p.456 - ↑
^{3.0}^{3.1}^{3.2}^{3.3}Serre (1979) p.162 - ↑ Gille & Szamuley (2006) p.142
- ↑ Gille & Szamuley (2006) p.143
- ↑ Gille & Szamuley (2006) p.144
- ↑
^{7.0}^{7.1}Fried & Jarden (2008) p.462 - ↑ Lorenz (2008) p.181
- ↑ Serre (1979) p.161
- ↑
^{10.0}^{10.1}^{10.2}Gille & Szamuely (2006) p.141 - ↑
^{11.0}^{11.1}Serre (1997) p.87 - ↑ Lang (1997) p.245
- ↑
^{13.0}^{13.1}{{#invoke:citation/CS1|citation |CitationClass=book }} - ↑ Lorenz (2008) p.116
- ↑ Lorenz (2008) p.119
- ↑
^{16.0}^{16.1}^{16.2}Serre (1997) p.88 - ↑
^{17.0}^{17.1}Fried & Jarden (2008) p.459 - ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Lang (1997) p.247
- ↑ Fried & Jarden (2008) p.457
- ↑
^{21.0}^{21.1}^{21.2}^{21.3}Fried & Jarden (2008) p.461

## References

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