On the corestriction homomorphism and generalized involutions of the second kind
Let K/F be a field extension of degree 2 and let A be a central simple K- algebra. A classical result of Albert says that A admits an involution of the second kind if and only if \(Cor_{K/F}([A])=1\). One of the main purposes of the paper is to extend this result to Galois extensions K/F of higher degree. Let \(\sigma_ 1=1,\sigma_ 2,...,\sigma_ n\) be the elements of G and let \(A_ i\) be the algebra A with the K-action twisted through \(\sigma_ i^{-1}\). A generalized involution of the second kind is an anti-imbedding \(A\to A_ n\otimes_ K...\otimes A_ 2\) with further properties. A generalized involution of the second kind exists for A if and only if \(Index\quad (Cor_{K/F}([A]))| (Index([A])^{n-2}).\) Nontrivial examples are split algebras and certain cyclic algebras. In the split case, (classical) involutions of the second kind arise from nondegenerate Hermitian forms. This construction is also generalized to arbitrary Galois extensions. The paper ends with some explicit examples. In particular, the author shows how his approach can be used to compute the corestriction of cyclic algebras of degree 3. This paper is a sequel to [ibid. 57, 449-465 (1979; Zbl 0408.16016) and 66, 205-219 (1980; Zbl 0451.16013)], where involutions of the first kind are generalized.
scientific article; zbMATH DE number 3889741