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, ..., 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.
- 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.
- 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.
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,
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 ( if no such number exists), is called the diophantine dimension dd(K) of K.
Every finite field is C1.
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 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
A field which is weakly Ck,d for every d is weakly Ck.
- 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 is weakly C1, then every field is weakly C1.
- Fried & Jarden (2008) p.455
- Fried & Jarden (2008) p.456
- Serre (1979) p.162
- Gille & Szamuley (2006) p.142
- Gille & Szamuley (2006) p.143
- Gille & Szamuley (2006) p.144
- Fried & Jarden (2008) p.462
- Lorenz (2008) p.181
- Serre (1979) p.161
- Gille & Szamuely (2006) p.141
- Serre (1997) p.87
- Lang (1997) p.245
- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 3-540-37888-X.
- Lorenz (2008) p.116
- Lorenz (2008) p.119
- Serre (1997) p.88
- Fried & Jarden (2008) p.459
- Terjanian, Guy (1966). "Un contre-example à une conjecture d'Artin". C. R. Acad. Sci. Paris Sér. A-B (in French). 262: A612. Zbl 0133.29705.
- Lang (1997) p.247
- Fried & Jarden (2008) p.457
- Fried & Jarden (2008) p.461
- Ax, James; Kochen, Simon (1965). "Diophantine problems over local fields I". Amer. J. Math. 87: 605–630. Zbl 0136.32805. doi:10.2307/2373065.
- Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 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. 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.
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- Tsen, C. (1936), "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper", J. Chinese Math. Soc., 171: 81–92, Zbl 0015.38803