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[parent] Jacobson's theorem on composition algebras (Theorem)

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$.

Bibliography

1
Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.



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See Also: composition algebras over $\mathbb{R}$, Hurwitz's theorem on composition algebras, composition algebra over algebaically closed fields, composition algebras over finite fields, composition algebras over $\mathbb{Q}$


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Cross-references: non-degenerate quadratic forms, dimensions, limits, signature, real numbers, index, isomorphism, algebra, isomorphic, characteristic, Cayley-Dickson algebras, unital, invertible linear map, isometric, quadratic form, field, composition algebra

This is version 1 of Jacobson's theorem on composition algebras, born on 2007-06-23.
Object id is 9651, canonical name is JacobsonsTheoremOnCompositionAlgebras.
Accessed 407 times total.

Classification:
AMS MSC17A75 (Nonassociative rings and algebras :: General nonassociative rings :: Composition algebras)

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