A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field [Formula: see text] can be reduced to the following three classifications, for each finite Galois extension [Formula: see text] of [Formula: see text]: (1) finite-dimensional central division algebras over [Formula: see text], up to isomorphism; (2) twisted group algebras of finite groups over [Formula: see text], up to graded-isomorphism; (3) [Formula: see text]-forms of certain graded matrix algebras with coefficients in [Formula: see text] where [Formula: see text] is as in (1) and [Formula: see text] is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.

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任意域上的分级除法代数

分级除法代数是由一个群分级的代数，使得所有非零齐次元素都是可逆的。这包括配备任意组分级（包括平凡分级）的除法代数。我们表明，对于 [公式：见文本] 的每个有限伽罗瓦扩展 [公式：见文本]，在任意域 [公式：见文本] 上的有限维分级中心分级划分代数的分类可以简化为以下三种分类：见正文]：（1）[公式：见正文]上的有限维中心除法代数，直至同构；(2) [公式：见正文]上有限群的扭曲群代数，直至分级同构；(3) [公式：见正文]——系数在 [公式：见正文] 中的某些分级矩阵代数的形式，其中 [公式：见正文] 如 (1) 和 [公式：见正文]如（2）。作为一个应用，我们将实数域（或任何实闭域）上的有限维分级除法代数分类为分级同构，并使用阿贝尔分级群。我们还讨论了字段的分组评分。