Decomposability of orthogonal involutions in degree 12

TL;DR

This paper extends Pfister's decomposition theorem from quadratic forms to orthogonal involutions on degree 12 algebras, providing criteria for trivial invariants and calculating the $f_3$-invariant.

Contribution

It generalizes Pfister's result to orthogonal involutions, showing they decompose into tensor products involving quaternion algebras, and offers criteria and invariants for such involutions.

Findings

Decomposition of degree 12 algebras with orthogonal involutions into tensor products.

Criteria for the existence of involutions with trivial invariants.

Calculation of the $f_3$-invariant for algebras of index 2.

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. 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 $2$.