Central simple algebra

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In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K (also called a Brauer algebra after Richard Brauer), is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K. In other words, any simple algebra is a central simple algebra over its center.

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.

According to the Artin–Wedderburn theorem a simple algebra A is isomorphic to 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.

[edit] Properties

• Every automorphism of a central simple algebra is an inner automorphism (follows from Skolem-Noether theorem)
• If S is a simple subalgebra of a central simple algebra A then dim_FS divides dim_FA
• Every 4 dimensional central simple algebra over a field F is isomorphic to a quaternion algebra; in fact, it is either a two-by-two matrix algebra, or a division algebra

[edit] See also

• Brauer group
• Severi-Brauer variety