Central simple algebra

In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A that is simple, and for which the center is exactly K.

For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Finite-dimensionality is essential to the definition: for instance, for a field F of characteristic 0, the Weyl algebra ${\displaystyle F[X,\partial _{X}]}$ is a simple algebra with center F, but is not a central simple algebra over F as it has infinite dimension as a F-module.

By the Artin–Wedderburn theorem, a finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion group.