A question on the discriminants of involutions of central division algebras
Let \(D\) be a finite-dimensional central simple algebra over a field \(k\) of characteristic not 2 with a \(k\)-linear involution \(\sigma\) of orthogonal type. As for quadratic forms, one can associate to \(\sigma\) a discriminant with values in \(k^ \times/{k^ \times}^ 2\). Assume that \(\dim D \geq 16\). If \(D\) is not a division algebra, it is easy to see that the set of discriminants of involutions of orthogonal type is the full group of reduced norms from \(D^ \times\) modulo squares in \(k^ \times\). In this beautiful short note, the authors show that this is also the case if \(D\) is a division algebra. Consequences are the (surprising) facts that the set of discriminants of involutions of orthogonal type is always a group and that \(D\) admits an involution of discriminant \(1\) (if it admits any). In fact the main result is that, for any \(D\) of dimension \(\geq 16\), whose class in the Brauer group is 2-torsion, every element in \(D\) is symmetric for an involution of orthogonal type whose discriminant is \(1\). The proof is in two steps: the case \(\dim D = 16\) and the reduction to this case. The case \(\dim D = 16\) was already known [see \textit{M.-A. Knus, T. Y. Lam, D. B. Shapiro} and \textit{J.-P. Tignol}: Discriminants of involutions on biquaternion algebras, in Am. Math. Soc. Proc. Santa Barbara (1992)], but the authors also give a very simple proof in this case.
finite-dimensional central simple algebra
scientific article; zbMATH DE number 537333
scientific article