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In mathematics, a field *F* is called **quasi-algebraically closed** (or ** C_{1}**) if every non-constant homogeneous polynomial

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*.

- 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 of dimension 1 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]}

- 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]}

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*^{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]}

Every finite field is *C*_{1}.^{[7]}

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 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*).

A field *K* is **weakly C_{k,d}** if for every homogeneous polynomial of degree

*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}**.

- 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 and the function fields are weakly
*C*_{1}, then every field is weakly*C*_{1}.^{[21]}

**^**Fried & Jarden (2008) p. 455- ^
^{a}^{b}^{c}^{d}Fried & Jarden (2008) p. 456 - ^
^{a}^{b}^{c}^{d}Serre (1979) p. 162 **^**Gille & Szamuley (2006) p. 142**^**Gille & Szamuley (2006) p. 143**^**Gille & Szamuley (2006) p. 144- ^
^{a}^{b}Fried & Jarden (2008) p. 462 **^**Lorenz (2008) p. 181**^**Serre (1979) p. 161- ^
^{a}^{b}^{c}Gille & Szamuely (2006) p. 141 - ^
^{a}^{b}Serre (1997) p. 87 **^**Lang (1997) p. 245- ^
^{a}^{b}Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008).*Cohomology of Number Fields*. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 978-3-540-37888-4. **^**Lorenz (2008) p. 116**^**Lorenz (2008) p. 119- ^
^{a}^{b}^{c}Serre (1997) p. 88 - ^
^{a}^{b}Fried & Jarden (2008) p. 459 **^**Terjanian, Guy (1966). "Un contre-example à une conjecture d'Artin".*Comptes Rendus de l'Académie des Sciences, Série A-B*(in French).**262**: A612. Zbl 0133.29705.**^**Lang (1997) p. 247**^**Fried & Jarden (2008) p. 457- ^
^{a}^{b}^{c}^{d}Fried & Jarden (2008) p. 461

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