Brauer group

Summary

The Brauer group of a field $K$ is the group of classes of central simple algebras over $K$ up to matrix algebras, see the definition. Its product is given by the tensor product. Each class contains exactly one central division ring up to isomorphism, so the group classifies the division algebras with center $K$ and finite dimension. It is isomorphic to the Galois cohomology group $H^2(K, (K^s)^\times)$, see Galois cohomology. Its computation for local and global fields is a main part of class field theory.