Graded-division algebras and Galois extensions

[摘要] Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras. On the other hand, given a finite abelian group G, any central simple G -graded division algebra over a field F is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of F and the isomorphism class of a GGalois extension of F. This connection is used to classify the simple G-Galois extensions of F in terms of a Galois field extension L/F with Galois group isomorphic to a quotient G/K and an element in the quotient Z(2)(K, Lx)/B-2(K, Fx) subject to certain conditions. Non-simple G-Galois extensions are induced from simple T-Galois extensions for a subgroup T of G. We also classify finite-dimensional G -graded-division algebras and, as an application, finite G -graded-division rings. (C) 2021 Elsevier B.V. All rights reserved.

[发布日期] 2021-12-01

[关键词] Graded-division algebra;Classification;Galois extension;Brauer group