  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.
