Vol. 304, No. 1, 2020

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Decomposability of orthogonal involutions in degree $12$

Anne Quéguiner-Mathieu and Jean-Pierre Tignol

Vol. 304 (2020), No. 1, 169–180
Abstract

A theorem of Pfister asserts that every 12-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from 2 decomposes as a tensor product of a binary quadratic form and a 6-dimensional quadratic form with trivial discriminant. Our main result extends Pfister’s result to orthogonal involutions: every central simple algebra of degree 12 with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree 6 with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree 12, and to calculate the f3-invariant of the involution if the algebra has index 2.

Keywords
algebra with involution, hermitian form, cohomological invariant
Mathematical Subject Classification 2010
Primary: 11E72
Secondary: 11E81, 16W10
Milestones
Received: 8 May 2019
Revised: 11 July 2019
Accepted: 25 July 2019
Published: 18 January 2020
Authors
Anne Quéguiner-Mathieu
LAGA – Institut Galilée
Université Paris 13
Villetaneuse
France
Jean-Pierre Tignol
ICTEAM
UCLouvain
Louvain-la-Neuve
Belgium