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SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

Part of: Rings and algebras arising under various constructions Division rings and semisimple Artin rings

Published online by Cambridge University Press:  08 April 2019

CHRISTIAN BROWN  and
SUSANNE PUMPLÜN Open the ORCID record for SUSANNE PUMPLÜN [Opens in a new window]
CHRISTIAN BROWN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom e-mails: christian_jb@hotmail.co.uk; susanne.pumpluen@nottingham.ac.uk
SUSANNE PUMPLÜN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom e-mails: christian_jb@hotmail.co.uk; susanne.pumpluen@nottingham.ac.uk
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Abstract

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

MSC classification

Primary: 16S35: Twisted and skew group rings, crossed products
Secondary: 16K20: Finite-dimensional
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Special Issue S1: Workshop on Nonassociative algebras and their applications , December 2020 , pp. S165 - S185
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

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