Jacobson's theorem on composition algebrasJacobsonsTheoremOnCompositionAlgebras2007-06-23 12:56:242007-06-23 12:56:24Theoremcomposition algebra\usepackage{latexsym}
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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 $x\in C$.
\begin{thm}[Jacobson]\cite[Theorem 3.23]{Schafer:nonass}
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.
\end{thm}
A Cayley-Dickson algebra is split if the algebra has non-trivial zero-divisors.
\begin{coro}\cite[Corollary 3.24]{Schafer:nonass}
Upto isomorphism there is only one split Cayley-Dickson algebra and the quadratic form
has Witt index 4.
\end{coro}
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 $8$.
\begin{thebibliography}{7}
\bibitem{Schafer:nonass}
Richard~D. Schafer, \emph{An introduction to nonassociative algebras}, Pure and
Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
\end{thebibliography}