Abstract
The aim of this article is to analyse a new field invariant, relevant to (formally) real fields, defined as the supremum of the dimensions of all anisotropic, weakly isotropic quadratic forms over the field. This invariant is compared with the classical u-invariant and with the Hasse number. Furthermore, in order to be able to obtain examples of fields where these invariants take certain prescribed values, totally positive field extensions are studied.
Similar content being viewed by others
References
Becher, K.J.: Supreme Pfister forms. Comm. Alg. 32, 217–241 (2004)
Elman, R., Lam, T.Y.: Quadratic forms and the u-invariant I. Math. Z. 131, 283–304 (1973)
Elman, R., Lam, T.Y., Wadsworth, A.R.: Orderings under field extensions. J. Reine Angew. Math. 306, 7–27 (1979)
Hoffmann, D.W.: Isotropy of quadratic forms and field invariants. Cont. Math. 272, 73–101 (2000)
Hoffmann, D.W.: Dimensions of Anisotropic Indefinite Quadratic Forms, I. Documenta Math., Quadratic Forms LSU 2001 183–200 (2001)
Lam, T.Y.: Introduction to quadratic forms over fields. Graduate Studies in Mathematics, 67. American Mathematical Society, Providence, RI, 2005.
Pfister, A.: Quadratic Forms with Applications to Algebraic Geometry and Topology. London Math. Soc. Lect. Notes 217. Cambridge University Press, 1995
Pierce, R.S.: Associative algebras. Graduate Texts in Mathematics 88, Springer- Verlag, New York, 1982
Prestel, A.: Remarks on the Pythagoras and Hasse number of real fields. J. Reine Angew. Math. 303/304, 284–294 (1978)
Prestel, A.: Lectures on Formally Real Fields. Lecture Notes in Math. 1093, Springer-Verlag, Berlin, 1984
Scharlau, W.: Quadratic and Hermitian forms. Grundlehren Math. Wiss. 270, Springer-Verlag, Berlin, 1985
Tignol, J.-P.: Réduction de l'indice d'une algèbre simple centrale sur le corps des fonctions d'une quadrique. Bull. Soc. Math. Belgique Sér. A 42, 735–745 (1990)