Jacobson’s theorem on composition algebras

Recall that composition algebraMathworldPlanetmath C over a field k is specified with a quadratic formMathworldPlanetmath q:Ck. Furthermore, two quadratic forms q:Ck and r:Dk are isometric if there exists an invertible linear mapMathworldPlanetmath f:CD such that r(f(x))=q(x) for all xC.

Theorem 1 (Jacobson).

[1, Theorem 3.23] Two unital Cayley-Dickson algebras C and D over a field k of characteristic not 2 are isomorphicPlanetmathPlanetmathPlanetmath if, and only if, their quadratic forms are isometric.

A Cayley-Dickson algebra is split if the algebraPlanetmathPlanetmathPlanetmath has non-trivial zero-divisors.

Corollary 2.

[1, Corollary 3.24] Upto isomorphismMathworldPlanetmathPlanetmath there is only one split Cayley-Dickson algebra and the quadratic form has Witt index 4.

Over the real numbers instead of Witt index, we say the signaturePlanetmathPlanetmath of the quadratic form is (4,4).

This result is often used together with a theorem of Hurwitz which limits the dimensionsPlanetmathPlanetmathPlanetmath of composition algebras to dimensions 1,2, 4 or 8. Thus to classify the composition algebras over a given field k of characteristic not 2, it suffices to classify the non-degenerate quadratic forms q:knk with n=1,2,4 or 8.


  • 1 Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
Title Jacobson’s theorem on composition algebras
Canonical name JacobsonsTheoremOnCompositionAlgebras
Date of creation 2013-03-22 17:18:14
Last modified on 2013-03-22 17:18:14
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 4
Author Algeboy (12884)
Entry type Theorem
Classification msc 17A75
Related topic CompositionAlgebrasOverMathbbR
Related topic HurwitzsTheoremOnCompositionAlgebras
Related topic CompositionAlgebraOverAlgebaicallyClosedFields
Related topic CompositionAlgebrasOverFiniteFields
Related topic CompositionAlgebrasOverMathbbQ