2019/11/05 by Quéguiner-Mathieu, Anne, Tignol, Jean-Pierre · 1 citation
#11E81 #FOS: Mathematics #K-Theory and Homology (math.KT) #Primary 11E72 #Secondary 16W10
paper · doi:10.48550/arxiv.1911.01782
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. The main result of the paper 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.