Geodesic
SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED
Glasgow mathematical journal, Tome 62 (2020), pp. S165-S185

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DOI

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. 
BROWN, CHRISTIAN; PUMPLÜN, SUSANNE. SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED. Glasgow mathematical journal, Tome 62 (2020), pp. S165-S185. doi: 10.1017/S0017089519000089
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     author = {BROWN, CHRISTIAN and PUMPL\"UN, SUSANNE},
     title = {SOLVABLE {CROSSED} {PRODUCT} {ALGEBRAS} {REVISITED}},
     journal = {Glasgow mathematical journal},
     pages = {S165--S185},
     year = {2020},
     volume = {62},
     number = {S1},
     doi = {10.1017/S0017089519000089},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000089/}
}
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