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.

中文翻译：

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重访可解交叉积代数

对于域上的任何中心简单代数F其中包含一个最大的子域米具有非平凡自同构群G= 自动F(米),G是可解的当且仅当代数包含子代数的有限链，这些子代数是在它们的中心上的广义循环代数（域扩展F) 满足一定条件。这些子代数与一个正常的子系列有关G. 叉积代数F因此，当且仅当它可以由这样一个有限的子代数链构成时，它是可解的。Petit 为除法叉积代数陈述了这一结果，并与 Albert 的类似结果重叠，然而，在这些术语中没有明确说明。特别是，每个可解的叉积除法代数都是一个广义循环代数F.