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 $\mathbb{F}$ is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of $\mathbb{F}$ and the isomorphism class of a G-Galois extension of $\mathbb{F}$.

This connection is used to classify the simple G-Galois extensions of $\mathbb{F}$ in terms of a Galois field extension $\mathbb{L}/\mathbb{F}$ with Galois group isomorphic to a quotient $G/K$ and an element in the quotient ${\mathrm{\text{Z}}}^2(K,{\mathbb{L}}^{\times })/{\mathrm{\text{B}}}^2(K,{\mathbb{F}}^{\times })$ 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.

梯度除法代数是由G组梯度化的有限维缔合代数理论的基础。如果G是阿贝尔阶，则可以使用循环构造将它们描述为中心简单的分度代数。

另一方面，给定一个有限的阿贝尔群G，一个域上的任何中心简单G阶除法代数$\mathbb{F}$ 归功于Picco和Platzeck的努力，是由（普通）Brauer组中的班级决定的 $\mathbb{F}$和G的-Galois扩展的同构类$\mathbb{F}$。

此连接用于分类简单的G -Galois扩展$\mathbb{F}$ 就Galois油田扩展而言 $\mathbb{大号}/\mathbb{F}$ Galois群同构为商 $G/ķ$ 和商中的一个元素 ${\mathrm{\text{ž}}}^{\mathrm{2个}}（ķ，{\mathbb{大号}}^{\times }）/{\mathrm{\text{乙}}}^{\mathrm{2个}}（ķ，{\mathbb{F}}^{\times }）$受某些条件约束。非简单的G - Galois扩展是从G的子集T的简单T - Galois扩展引起的。我们还对有限维G级划分代数和有限G级划分环进行了分类。