On division algebras of degree 3 with involution
type: Item
instance of: scholarly article
title: On division algebras of degree 3 with involution
zbMATH Open document ID: 0861.16012
author: Darrell Haile
author: M.-A. Knus
published in: Journal of Algebra
publication date: 1997-04-20T00:00:00Z
review text: The authors present a simple proof of the theorem [\textit{A. A. Albert}, Scripta Math. 26, 309-316 (1963; Zbl 0147.28702)] that a division algebra of degree 3 (in characteristic not 3) with an involution \(J\) of the second kind has a maximal subfield which is Galois over the fixed field of \(J\), with the symmetric group of degree 3 as Galois group. The argument also provided another proof of Wedderburn's theorem that central division algebras of degree 3 are cyclic. The result is applied to prove that every central simple algebra of degree 3 with an involution of the second kind is a homomorphic image of the Clifford algebra of a binary cubic form with its canonical involution. For a given form this leads to a classification of these images.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 941280
zbMATH Keywords: division algebras of degree 3
zbMATH Keywords: involutions
zbMATH Keywords: maximal subfields
zbMATH Keywords: Galois groups
zbMATH Keywords: Wedderburn's theorem
zbMATH Keywords: central division algebras
zbMATH Keywords: central simple algebras
zbMATH Keywords: Clifford algebras
zbMATH Keywords: binary cubic forms
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
OpenAlex ID: W2056860440
DOI: 10.1006/JABR.1996.0299
description: scientific article; zbMATH DE number 941280