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 Free Access

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] Arnaud Beauville, Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Peter Swinnerton-Dyer, Variétés stablement rationnelles non rationnelles, Ann. of Math. (2) 121 (1985), no. 2, 283–318 (French). MR 786350, https://doi.org/10.2307/1971174
[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] Manfred Knebusch, Generic splitting of quadratic forms. II, Proc. London Math. Soc. (3) 34 (1977), no. 1, 1–31. MR 0427345, https://doi.org/10.1112/plms/s3-34.1.1
[L] T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980. Revised second printing; Mathematics Lecture Note Series. MR 634798
[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] Jack Ohm, On subfields of rational function fields, Arch. Math. (Basel) 42 (1984), no. 2, 136–138. MR 751484, https://doi.org/10.1007/BF01772932
[O2] Jack Ohm, On ruled fields, Sém. Théor. Nombres Bordeaux (2) 1 (1989), no. 1, 27–49 (English, with French summary). MR 1050263
[O3] Jack Ohm, Function fields of conics, a theorem of Amitsur-MacRae, and a problem of Zariski, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 333–363. MR 1272040
[W] Adrian R. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields, Trans. Amer. Math. Soc. 208 (1975), 352–358. MR 376527, https://doi.org/10.1090/S0002-9947-1975-0376527-8