A note on trace forms and involutions of the first kind
Let \(A\) be a central simple algebra over a field \(F\) of characteristic different from \(2\), and let \(\sigma\) be an involution of the first kind on \(A\), i.e., an anti-automorphism of period \(2\) which is the identity on \(F\). The involution trace form \(\phi^{\sigma}\colon A\to F\) is defined by \(\phi^{\sigma}(x)=\text{tr}(x\sigma(x))\) where \(\text{tr}\) is the reduced trace. The author computes the Hasse invariant of \(\phi^{\sigma}\) and the discriminant of its restriction \(\phi^{\sigma}_{1}\) to the space of symmetric elements. It turns out that the discriminant of \(\phi^{\sigma}_{1}\) is directly related to the discriminant of the involution \(\sigma\), as defined by \textit{M. A. Knus, R. Parimala} and \textit{R. Sridharan} [Bull. Soc. Math. Belg., Sér. A 43, No. 1/2, 89-98 (1991; Zbl 0757.12002)]. The Hasse invariant of \(\phi^{\sigma}\) depends only on the Brauer class and the degree of \(A\) and on the discriminant of \(\sigma\). The same results have been obtained by \textit{A. Quéguiner} [``Cohomological invariants of algebras with involution'', J. Algebra 194, No. 1, 299-330 (1997)].
scientific article; zbMATH DE number 1051310