composition algebra over algebaically closed fields

Theorem 1.

There are 4 non-isomorphic composition algebrasMathworldPlanetmath over an algebraically closed field k: one division algebra, the field itself, and the three split algebrasPlanetmathPlanetmath.

  1. 1.


  2. 2.

    The exchange algebra: kk.

  3. 3.

    2×2 matrices over k: M2(k).

  4. 4.

    The cross-product of 2×2-matrices over k: M2(k)M2(k).


To see this recall that every composition algebra comes equipped with a quadratic formMathworldPlanetmath. Any 2-dimensional anisotropic subspace arises from a quadratic field extension. As our field is algebraically closedMathworldPlanetmath the quadratic form has no anisotropic subspaces and is therefore the unique quadratic form of maximal Witt index. Following Hurwitz’s theorem we know the composition algebras come in dimensions 1,2,4, and 8 and arise by the Cayley-Dickson method. Thus we have the field itself and the three split composition algebras. ∎

Title composition algebra over algebaically closed fields
Canonical name CompositionAlgebraOverAlgebaicallyClosedFields
Date of creation 2013-03-22 17:18:22
Last modified on 2013-03-22 17:18:22
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 6
Author Algeboy (12884)
Entry type Theorem
Classification msc 17A75
Related topic HurwitzsTheorem
Related topic JacobsonsTheoremOnCompositionAlgebras