Discriminant and Clifford algebras
Let \(F\) be a field of characteristic not equal to \(2\), and let \(A\) be a central simple \(F\)-algebra of degree \(n=4m\), for some positive integer \(m\), endowed with an orthogonal involution \(\sigma\). An element \(\theta\in A\) is said to be a square-central skew-symmetric unit, if \(\sigma(\theta)=-\theta\) and \(\theta^2\in F^*\). It is known that if \(\theta\) is such a unit, then \(\sigma\) induces on the centralizer \(\widetilde A=C_A(\theta)\) a unitary involution \(\widetilde\sigma\), and \(\widetilde A\) is a central simple algebra of degree \(2m\) over the etale quadratic extension \(F(\theta)\) of \(F\); also, one can associate with these data two different central simple algebras with involutions, namely, the Clifford algebra \(C(A,\sigma)\) of \((A,\sigma)\), and the discriminant algebra \(D(\widetilde A,\widetilde\sigma)\) of \((\widetilde A,\widetilde\sigma)\), both endowed with their canonical involutions [see \textit{M.-A. Knus, A. S. Merkurjev, M. Rost} and \textit{J.-P. Tignol}, The book of involutions, Colloq. Publ. Am. Math. Soc. 44, Providence, RI (1998; Zbl 0955.16001)]. The paper under review concentrates on the study of the special case \(m=2\). It shows that then \(A\) contains a square-central skew-symmetric unit if and only if the discriminant of \(\sigma\) is trivial, so \(C(A,\sigma)=C_1\times C_2\), for some central simple \(F\)-algebras \(C_1\) and \(C_2\) of degree \(8\), such that at least one of them is of index \(1\) or \(2\). As pointed out by the authors, this result enables one to find without difficulty examples of degree \(8\) central simple algebras with orthogonal involutions that do not contain such a unit. The main theorem of the paper states (for \(m=2\)) that \((A,\sigma)\) is decomposable, i.e. \((A,\sigma)\cong(H,\sigma_H)\otimes_F(A_0,\sigma_0)\) where \(H\) is a quaternion algebra, \(A_0\) is of degree \(4\) and \(\sigma=\sigma_H\otimes\sigma_0\), if and only if the discriminant of \(\sigma\) is trivial, so \(C(A,\sigma)=C_1\times C_2\), for some central simple \(F\)-algebras \(C_1\), \(C_2\) of degree \(8\), at least one of which contains a square symmetric unit whose centralizer is split. The proof of the main theorem is based on several results of independent interest (covering the case of an arbitrary \(m\in\mathbb{N}\)). In particular, this applies to the one that the algebra \(D(\widetilde A,\widetilde\sigma)\) is an orthogonal summand in one of the components of \(C(A,\sigma)\).
scientific article; zbMATH DE number 1801562