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.

Read more about this topic:  Central Simple Algebra

Famous quotes containing the word properties:

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)