Solvable crossed product algebras revisited

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.

References in corpus (1)

• Finite nonassociative algebras obtained from skew polynomials and possible applications to $(f,\sigma ,\delta )$-codes

Cited by in corpus (1)

• Nonassociative cyclic extensions of fields and central simple algebras