# Jacobson’s theorem on composition algebras

Recall that composition algebra  $C$ over a field $k$ is specified with a quadratic form  $q:C\to k$. Furthermore, two quadratic forms $q:C\to k$ and $r:D\to k$ are isometric if there exists an invertible linear map  $f:C\to D$ such that $r(f(x))=q(x)$ for all $x\in C$.

###### Theorem 1 (Jacobson).

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

This result is often used together with a theorem of Hurwitz which limits the dimensions   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 $q:k^{n}\to k$ with $n=1,2,4$ or $8$.

## References

• 1 Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
Title Jacobson’s theorem on composition algebras JacobsonsTheoremOnCompositionAlgebras 2013-03-22 17:18:14 2013-03-22 17:18:14 Algeboy (12884) Algeboy (12884) 4 Algeboy (12884) Theorem msc 17A75 CompositionAlgebrasOverMathbbR HurwitzsTheoremOnCompositionAlgebras CompositionAlgebraOverAlgebaicallyClosedFields CompositionAlgebrasOverFiniteFields CompositionAlgebrasOverMathbbQ