Solvable Crossed Product Algebras Revisited

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, is not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F .

Nonassociative cyclic extensions of fields and central simple algebras

“…If G is solvable then A is called a solvable G-crossed product. In [9] we revisited a result by Albert [1] on solvable crossed products and gave a proof for Albert's result using generalized cyclic algebras following Petit's approach [17], proving that a G-crossed product is solvable if and only if it can be constructed as a chain of generalized cyclic algebras. Hence any solvable G-crossed product division algebra is always a generalized cyclic division algebra.…”