Decomposability of orthogonal involutions in degree 12 - Archive ouverte HAL
Article Dans Une Revue Pacific J. Math. Année : 2020

Decomposability of orthogonal involutions in degree 12

Résumé

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 $f_3$-invariant of the involution if the algebra has index $2$.

Dates et versions

hal-03858859 , version 1 (17-11-2022)

Identifiants

Citer

Anne Quéguiner-Mathieu, Jean-Pierre Tignol. Decomposability of orthogonal involutions in degree 12. Pacific J. Math., 2020, 304 (1), pp.169-180. ⟨hal-03858859⟩
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