On the corestriction homomorphism and generalized involutions of the second kind
type: Item
instance of: scholarly article
title: On the corestriction homomorphism and generalized involutions of the second kind
zbMATH Open document ID: 0558.16011
DOI: 10.1016/0021-8693(84)90109-1
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 3889741
zbMATH Keywords: central simple K-algebra
zbMATH Keywords: involution of the second kind
zbMATH Keywords: Galois extensions
zbMATH Keywords: generalized involution
zbMATH Keywords: split algebras
zbMATH Keywords: cyclic algebras
zbMATH Keywords: Hermitian forms
zbMATH Keywords: corestriction
author: Darrell Haile
reviewed by: M.-A. Knus
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90109-1
OpenAlex ID: W2013544157
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description: scientific article; zbMATH DE number 3889741