Bulletin of the Belgian Mathematical Society - Simon Stevin

The orthogonal $u$-invariant of a quaternion algebra

Karim Johannes Becher and Mohammad G. Mahmoudi

Full-text: Open access


In quadratic form theory over fields, a much studied field invariant is the $u$-invariant, defined as the supremum of the dimensions of anisotropic quadratic forms over the field. We investigate the corresponding notions of $u$-invariant for hermitian and for skew-hermitian forms over a division algebra with involution, with a special focus on skew-hermitian forms over a quaternion algebra with canonical involution. Under certain conditions on the center of the quaternion algebra, we obtain sharp bounds for this invariant.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 1 (2010), 181-192.

First available in Project Euclid: 5 March 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11E04: Quadratic forms over general fields 11E39: Bilinear and Hermitian forms 11E81: Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24]

hermitian form involution division algebra isotropy system of quadratic forms discriminant Tsen-Lang Theory Kneser's Theorem local field Kaplansky field


Becher, Karim Johannes; Mahmoudi, Mohammad G. The orthogonal $u$-invariant of a quaternion algebra. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 1, 181--192. doi:10.36045/bbms/1267798507. https://projecteuclid.org/euclid.bbms/1267798507

Export citation