Central Simple Algebra - Properties

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.

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