Central simple algebra
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In ring theory and related areas of mathematics a central simple algebra (CSA) over K, also called a Brauer algebra after Richard Brauer, is a finite-dimensional (associative) algebra A, which is a simple ring, and for which the center is exactly K. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large).
According to Artin–Wedderburn theorem a simple algebra A is isomorphic to M(n,S) for some skew field 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 if their skew fields S and T are isomorphic. The of set all equivalence classes of central simple algebras over a given field F can be equipped with a group operation and is called the Brauer group Br(F).
Examples
- the quaternion and more generally any quaternion algebra
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 dimFS divides dimFA
- Every 4 dimensional central simple algebra over a field F with char(F) ≠ 2 is isomorpic to a quaternion algebra
