Properties
- There is a unique division algebra in each Brauer equivalence class.
- Every automorphism of a central simple algebra is an inner automorphism (follows from 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 CSA, or of a class, is the degree of the equivalent division algebra.
- If S is a simple subalgebra of a central simple algebra A then dimF S divides dimF A.
- 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.
Read more about this topic: Central Simple Algebra
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