Brauer group

Over a field K there are many central simple algebras, but many of them differ only by harmless matrix padding — M_2(D) and M_3(D) are built from the same underlying division algebra D and ought to count as the same thing. The Brauer group is what you get by declaring two central simple algebras equivalent when they share the same underlying division algebra, then assembling the equivalence classes into a group whose operation is tensor product. It packages all the noncommutative division algebras over K into one abelian group.

Precisely, two central simple K-algebras A and B are Brauer equivalent if M_r(A) is isomorphic to M_s(B) for some r, s; equivalently they have isomorphic underlying division algebras. The set Br(K) of equivalence classes is an abelian group: the product of the classes of A and B is the class of A tensor B over K, the identity is the class of K itself (all split matrix algebras), and the inverse of the class of A is the class of the opposite algebra A-op, since A tensor A-op is a matrix algebra.

Each Brauer class has a unique division-algebra representative, so Br(K) literally enumerates the central division algebras over K up to isomorphism. The order of a class in the group is called its period, the degree of its division-algebra representative is its index, and a celebrated theorem says the period always divides the index and shares the same prime factors.

Computing Br(K) is the heart of arithmetic. For algebraically closed K the group is trivial; for R it is Z/2Z generated by the quaternions; for a local field it is canonically Q/Z; and the Albert-Brauer-Hasse-Noether theorem gives the exact sequence for a global field, a cornerstone of class field theory. Cohomologically Br(K) is identified with the Galois cohomology group H^2 of the absolute Galois group acting on the multiplicative group of a separable closure.