Hurwitz’s theorem on composition algebras


Theorem 1 (Hurwitz).

[1, Theorem 3.25] Given a field k of characteristic not 2, then every unital composition algebraMathworldPlanetmath C over k is isomorphic to one of:

  1. 1.

    k,

  2. 2.

    (αk) for α∈k,

  3. 3.

    (α,βk) for α,β∈k,

  4. 4.

    (α,β,γk) for α,β,γ∈k.

In particular, all composition algebras over k are finite dimensional and of dimensionMathworldPlanetmath 1, 2, 4 or 8.

References

  • 1 Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
Title Hurwitz’s theorem on composition algebras
Canonical name HurwitzsTheoremOnCompositionAlgebras
Date of creation 2013-03-22 17:18:20
Last modified on 2013-03-22 17:18:20
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 4
Author Algeboy (12884)
Entry type Theorem
Classification msc 17A75
Related topic JacobsonsTheoremOnCompositionAlgebras