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arXiv · 1702.04605

Solvable crossed product algebras revisited

Christian Brown·Susanne Pumpluen
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Abstract

For any central simple algebra over a field F which contains a maximal subfield M with non-trivial F-automorphism group G, 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, is not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

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Christian Brown, Susanne Pumpluen. 2017-02-15. Solvable crossed product algebras revisited. https://doi.org/10.1017/s0017089519000089

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