# Solvable crossed product algebras revisited

## TL;DR

This paper characterizes solvable crossed product algebras over a field F by linking their structure to chains of generalized cyclic subalgebras, extending previous results by Petit and Albert.

## Contribution

It provides a new characterization of solvable crossed product algebras using chains of subalgebras related to automorphism groups, clarifying their construction.

## Key findings

- Solvable crossed product algebras contain chains of generalized cyclic subalgebras.
- Every solvable crossed product division algebra is a generalized cyclic algebra.
- The results extend and clarify earlier work by Petit and Albert.

## 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.