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SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

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

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: [email protected]; [email protected]
SUSANNE PUMPLÜN
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom e-mails: [email protected]; [email protected]
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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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References

Albert, A. A., Structure of algebras. Revised printing. American Mathematical Society Colloquium Publications, vol. XXIV (American Mathematical Society, Providence, RI, 1961).Google Scholar
Amitsur, A. S., Non-commutative cyclic fields, Duke Math. J. 21 (1954), 87–105.Google Scholar
Amitsur, A. S. and Saltman, D. J., Generic abelian crossed products and p-algebras, J. Alg. 51 (1978), 76–87.CrossRefGoogle Scholar
Berhuy, G. and Oggier, F., An introduction to central simple algebras and their applications to wireless communication, in Mathematical surveys and monographs, 191 (American Mathematical Society, Providence, RI, 2013), viii+276 pp.CrossRefGoogle Scholar
Brown, C., Petit algebras and their automophisms, PhD Thesis (University of Nottingham, Nottingham, UK, 2018). arXiv:1806.00822v1 [math.RA]Google Scholar
Hanke, T., A direct approach to noncrossed product division algebras, PhD Thesis (Universität Potsdam, Potsdam, Germany, 2011). arXiv:1109.1580v1 [math.RA]Google Scholar
Jacobson, N., Finite-dimensional division algebras over fields (Springer Verlag, Berlin, Heidelberg, New York, 1996).CrossRefGoogle Scholar
Keshavarzipour, T. and Mahdavi-Hezavehi, M., Crossed product conditions for central simple algebras in terms of irreducible subgroups, J. Algebra 315(2) (2007), 738–744.CrossRefGoogle Scholar
Kiani, D. and Mahdavi-Hezavehi, M., Crossed product conditions for division algebras of prime power degree, J. Alg. 283 (2005), 222–231.CrossRefGoogle Scholar
Kursov, V. V. and Yanchevskii, V. I., Crossed products of simple algebras and their automorphism groups, Amer. Math. Soc. Transl. 154(2) (1992), 75–80.CrossRefGoogle Scholar
Motiee, M., A note on the existence of cyclic algebras in division algebras, Comm. Alg. 45(10) (2017), 4396–4399.CrossRefGoogle Scholar
Petit, J.-C., Sur certains quasi-corps généralisant un type d’anneau-quotient, Séminaire Dubriel. Algèbre et théorie des nombres 20 (1966–1967), 1–18.Google Scholar
Petit, J.-C., Sur les quasi-corps distributifes à base momogène, C. R. Acad. Sc. Paris, Série A 266 (1968), 402–404.Google Scholar
Pumplün, S., Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, σ) -codes, Adv. Math. Commun. 11(3) (2017), 615–634.Google Scholar
Schacher, M. M., Subfields of division rings, I, J. Algebra 9(4) (1968), 451–477.CrossRefGoogle Scholar
Sonn, J., ℚ-admissibility of solvable groups, J. Algebra 84(2) (1983), 411–419.CrossRefGoogle Scholar
Teichmüller, O., Über die sogenannte nichtkommutative Galoissche Theorie und die Relation ξλ, μ, νξλ, μν, πε, λμ, ν, π = πε, λμ, ν, π (German) Deutsche Math. 5 (1940), 138–149.Google Scholar
Tignol, J.-P., Generalized crossed products, in Séminaire Mathématique (nouvelle série), vol. 106 (Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 1987).Google Scholar