Graded-division algebras and Galois extensions |
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Institution: | 1. Departamento de Matemáticas e Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain;2. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John''s, NL, A1C5S7, Canada |
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Abstract: | 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 is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of and the isomorphism class of a G-Galois extension of .This connection is used to classify the simple G-Galois extensions of in terms of a Galois field extension with Galois group isomorphic to a quotient and an element in the quotient 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. |
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Keywords: | Graded-division algebra Classification Galois extension Brauer group |
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