The Zariski problem for function fields of quadratic forms

Ohm, Jack

doi:10.1090/S0002-9939-96-03238-8

The Zariski problem for function fields of quadratic forms

Author: Jack Ohm
Journal: Proc. Amer. Math. Soc. 124 (1996), 1679-1685
MSC (1991): Primary 11E04, 11E81, 12F20
DOI: https://doi.org/10.1090/S0002-9939-96-03238-8
MathSciNet review: 1307553
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By `a quadratic function field' is meant the affine function field of a nonsingular quadratic form of dimension $> 2$ . What quadratic function fields contain a given quadratic function field $k(P)$ ? This problem is solved here for quadratic forms $P$ of dimensions 3 and 4, and an application to the Zariski cancellation problem for quadratic function fields is given.

[AO] H. Ahmad and J. Ohm, Function fields of Pfister neighbors, J. Algebra 178 (1995), 653--664.
[B] A. Beauville, J.-L. Colliot-Th $\acute e$ l $\grave e$ ne, J.-J. Sansuc, and Sir Peter Swinnerton-Dyer, Varietes stablement rationnelles non rationnelles, Annals of Math. 121 (1986), 283--318. MR 86m:14009
[H1] D. Hoffmann, Isotropy of 5-dimensional Quadratic forms over the function field of a quadric, preprint.
[H2] ------, On 6-dimensional quadratic forms isotropic over the function field of a quadric, Comm. in Algebra 22(6) (1994), 1999-2014. CMP 94:10
[H3] ------, Minimal quadratic forms and function fields of quadratic forms, preprint.
[K] M. Knebusch, Generic splitting of quadratic forms I, Proc. London Math. Soc. 33 (1976), 65-93; II Proc. London Math. Soc., vol. 34, 1977, pp. 1--31. MR 55:379
[L] T.Y. Lam, The algebraic theory of quadratic forms, Benjamin, Reading, Mass., 1980. MR 83d:10022
[L2] ------, Fields of u-invariant 6 after A. Merkurjev, Isarel Math. Conf. Proc., Ring Theory 1989 (in honor of S.A. Amitsur) (L. Rowen, ed.), vol. I, Weizmann Science Press, Jerusalem, 1989, pp. 12--31.
[O1] J. Ohm, On subfields of rational function fields, Arch. Math 42 (1984), 136-138. MR 86h:12007
[O2] ------, On ruled fields, Sem. de theorie des nombres, Bordeaux 1, Serie 2 1 (1989), 27-50. MR 91g:12006
[O3] ------, Function fields of conics, a theorem of Amitsur-MacRae, and a problem of Zariski, Algebraic geometry and its applications (C. Bajaj, ed.), Springer-Verlag, 1994, pp. 333--363. MR 95c:12010
[W] A. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields, Trans. Amer. Math. Soc. 208 (1975), 352--358. MR 51:12702