# 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.

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

###### Corollary 2.

[1, Corollary 3.24] Upto isomorphism 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 signature of the quadratic form is $(4,4)$.

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.
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