Graded-division algebras over arbitrary fields

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 ? can be reduced to the following three classifications, for each finite Galois extension ? of ?: (1) finite-dimensional central division algebras over ?, up to isomorphism; (2) twisted group algebras of finite groups over ?, up to graded-isomorphism; (3) ?-forms of certain graded matrix algebras with coefficients in Delta circle times?? where Delta is as in (1) and ? 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.