Quasi-algebraically closed field

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 (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

X₁, ..., 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(x₁, ..., 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.^[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]

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 C₁. 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 C₀ 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 (${\displaystyle \infty }$ if no such number exists), is called the diophantine dimension dd(K) of K.^[13]

C₁ fields

Every finite field is C₁.^[7]

C₂ fields

Properties

Suppose that the field k is C₂.

• 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 C₂, 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 Pⁿ(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.^[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 C₁.^[21]
• Any field with procyclic absolute Galois group is weakly C₁.^[21]
• Any field of positive characteristic is weakly C₂.^[21]
• If the field of rational numbers ${\displaystyle \mathbb {Q} }$ and the function fields ${\displaystyle \mathbb {F} _{p}(t)}$ are weakly C₁, then every field is weakly C₁.^[21]

See also

• Brauer's theorem on forms
• Tsen rank

Citations

References

• Ax, James; Kochen, Simon (1965). "Diophantine problems over local fields I". Amer. J. Math. 87 (3): 605–630. doi:10.2307/2373065. JSTOR 2373065. Zbl 0136.32805.
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
• Greenberg, M.J. (1969). Lectures of forms in many variables. Mathematics Lecture Note Series. New York-Amsterdam: W.A. Benjamin. Zbl 0185.08304.
• Lang, Serge (1952), "On quasi algebraic closure", Annals of Mathematics, 55 (2): 373–390, doi:10.2307/1969785, JSTOR 1969785, Zbl 0046.26202
• Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. pp. 109–126. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. Vol. 67. Translated by Greenberg, Marvin Jay. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016.
• Serre, Jean-Pierre (1997). Galois cohomology. Springer-Verlag. ISBN 3-540-61990-9. Zbl 0902.12004.
• Tsen, C. (1936), "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper", J. Chinese Math. Soc., 171: 81–92, Zbl 0015.38803