Abstract
For a field F of characteristic not 2, let \({\widehat{F}}\) denote the maximal dimension of anisotropic, weakly isotropic, non-degenerate quadratic forms over F. In this paper, we investigate the behavior of this invariant under field extensions.
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References
Becher K.J.: Minimal weakly isotropic forms. Math. Z. 252(1), 91–102 (2006) MR 2209153 (2006k:11060)
Becher K.J.: Totally positive extensions and weakly isotropic forms. Manuscripta Math. 120(1), 83–90 (2006) MR 2223482
Bos, R.: Quadratic forms, orderings and abstract Witt rings. Dissertation, Rijksuniversiteit te Utrecht, Utrecht (1984). With a Dutch summary. MR 756809 (86h:11033)
Bröcker L.: Zur Theorie der quadratischen Formen über formal reellen Körpern. Math. Ann. 210, 233–256 (1974) MR 0354549 (50 #7027)
Elman, R., Karpenko, N., Merkurjev, A.: The algebraic and geometric theory of quadratic forms. American Mathematical Society Colloquium Publications, vol. 56. American Mathematical Society, Providence, RI (2008). MR 2427530 (2009d:11062)
Elman R., Lam T.Y.: Quadratic forms under algebraic extensions. Math. Ann. 219(1), 21–42 (1976) MR 0401649 (53 #5476)
Elman, R.: Quadratic forms and the u-invariant. In: III, Conference on Quadratic Forms—1976 (Proceedings of Conference Queen’s University, Kingston, Ontario, 1976), Queen’s University, Kingston, Ontario, 1977, pp. 422–444. Queen’s Papers in Pure and Applied Mathematics, No. 46. MR 0491490 (58 #10732)
Elman R., Lam T.Y., Prestel A.: On some Hasse principles over formally real fields. Math. Z. 134, 291–301 (1973) MR 0330045 (48 #8384)
Elman R., Prestel A.: Reduced stability of the Witt ring of a field and its Pythagorean closure. Am. J. Math. 106(5), 1237–1260 (1984) MR 761585 (86f:11030)
Hoffmann, D.W.: Dimensions of anisotropic indefinite quadratic forms. In: I, Proceedings of the Conference on Quadratic Forms and Related Topics (Baton Rouge, LA, 2001), no. Extra vol., 2001, pp. 183–200 (electronic). MR 1869394 (2002j:11032)
Lam, T.Y.: Introduction to quadratic forms over fields. Graduate Studies in Mathematics, vol. 67. American Mathematical Society, Providence, RI (2005). MR 2104929 (2005h:11075)
Marshall M.A.: Spaces of orderings and abstract real spectra Lecture Notes in Mathematics. vol. 1636. Springer, Berlin (1996)
Meißner W.: Quadratische Räume unendlicher Dimension und partielle Isomorphismen. Math. Z. 181(4), 481–495 (1982) MR 682669 (84i:10020)
Prestel A.: Remarks on the Pythagoras and Hasse number of real fields. J. Reine Angew. Math. 303(304), 284–294 (1978) MR 514686 (80d:12020)
Prestel A.: Lectures on Formally Real Fields. Lecture Notes in Mathematics. vol. 1093. Springer, Berlin (1984)
Prestel A., Ware R.: Almost isotropic quadratic forms. J. Lond. Math. Soc. (2) 19(2), 241–244 (1979) MR 533322 (80e:12024)
Schubert C.: Going up of the u-invariant over formally real fields. Math. Ann. 338(3), 757–765 (2007) MR 2317936 (2008d:11031)