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 which is simple, and for which the center is exactly K. (Note that not every simple algebra is a central simple algebra over its center: for instance, if K is a field of characteristic 0, then the Weyl algebra is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.) 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). 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. According to the Artin–Wedderburn theorem a finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Hence, there is a unique division algebra in each Brauer equivalence class. Every automorphism of a central simple algebra is an inner automorphism (this follows from the Skolem–Noether theorem). The dimension of a central simple algebra as a vector space over its centre is always a square: the degree is the square root of this dimension. The Schur index of a central simple algebra is the degree of the equivalent division algebra: it depends only on the Brauer class of the algebra. The period or exponent of a central simple algebra is the order of its Brauer class as an element of the Brauer group. It is a divisor of the index, and the two numbers are composed of the same prime factors.