The group is named for the algebraist Richard Brauer.

In order to define the Brauer group, one calls a **central simple algebra** (**CSA**) over *K* a finite-dimensional (associative) algebra *K* which is a simple ring, and for which the center is exactly *K*. That is, for the example of the real numbers just given, we do want to exclude the complex numbers **C**, which has larger center.

Given two such central simple algebras *A* and *B*, one defines a product on

- .

Given this closure property for CSAs, they form a monoid under tensor product. To get a group, apply the Artin-Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as M(*n*,D), an *n*x*n* matrix ring over a division algebra D. If we look just at D, rather than the value of *n*, the monoid becomes a group. That is, if we impose an equivalence relation identifying M(*m*,D) with M(*n*,D) for all integers *m* and *n* at least 1, we get a congruence relation; and the congruence classes are all invertible.

In the further theory, the Brauer groups of local fields are computed (they all turn out to be subgroups of **Q**/**Z**, for p-adic fields); and the results applied to global fields. This gives one approach to class field theory. It also has been applied to Diophantine equations.

In the general theory the Brauer group is expressed by factor sets; and expressed in terms of Galois cohomology. A generalisation via the theory of Azumaya algebras was introduced in algebraic geometry by Grothendieck.