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, ..., 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 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.
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 ( ∞ {\displaystyle \infty } if no such number exists), is called the diophantine dimension dd(K) of K.
C1 fields Every finite field is C1.
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. 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 that 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 Q {\displaystyle \mathbb {Q} } and the function fields F p ( t ) {\displaystyle \mathbb {F} _{p}(t)} are weakly C1, then every field is weakly C1.
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
