Decomposability of orthogonal involutions in degree $12$

Anne Quéguiner-Mathieu and Jean-Pierre Tignol

Vol. 304 (2020), No. 1, 169–180

DOI: 10.2140/pjm.2020.304.169

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 $f_3$-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