Graded-division algebras and Galois extensions

TL;DR

This paper classifies finite-dimensional graded-division algebras and Galois extensions over fields, revealing their structure through Brauer groups, Galois theory, and cocycle conditions, especially for abelian grading groups.

Contribution

It provides a classification of simple G-Galois extensions and graded-division algebras using Brauer groups and Galois cohomology, extending understanding of their structure.

Findings

Classification of simple G-Galois extensions via Galois groups and cocycles

Description of graded-division algebras through Brauer group elements

Extension of classification to non-simple G-Galois extensions

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 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 G-Galois 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 the class of a 2-cocycle of K with values in the multiplicative group of L modulo a 2-coboundary with values in the multiplicative group of F, 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.