Hurwitz’s theorem on composition algebras

[1, Theorem 3.25]Given a field $k$ of characteristic not $2$ , then every unital composition algebra $C$ over $k$ is isomorphic to one of:

1.
$k$ ,
2.
$\\left(\\frac{\\alpha}{k}\\right)$ for $\\alpha\\in k$ ,
3.
$\\left(\\frac{\\alpha,\\beta}{k}\\right)$ for $\\alpha,\\beta\\in k$ ,
4.
$\\left(\\frac{\\alpha,\\beta,\\gamma}{k}\\right)$ for $\\alpha,\\beta,\\gamma\\in k$ .

In particular, all composition algebras over $k$ are finite dimensional and of dimension $1$ , $2$ , $4$ or $8$ .

References

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

单词	HurwitzsTheoremOnCompositionAlgebras
释义	Hurwitz’s theorem on composition algebras Theorem 1 (Hurwitz). [1, Theorem 3.25]Given a field $k$ of characteristic not $2$ , then every unital composition algebra $C$ over $k$ is isomorphic to one of: 1. $k$ , 2. $\\left(\\frac{\\alpha}{k}\\right)$ for $\\alpha\\in k$ , 3. $\\left(\\frac{\\alpha,\\beta}{k}\\right)$ for $\\alpha,\\beta\\in k$ , 4. $\\left(\\frac{\\alpha,\\beta,\\gamma}{k}\\right)$ for $\\alpha,\\beta,\\gamma\\in k$ . In particular, all composition algebras over $k$ are finite dimensional and of dimension $1$ , $2$ , $4$ or $8$ . References 1 Richard D. Schafer, An introduction to nonassociative algebras, Pure andApplied Mathematics, Vol. 22, Academic Press, New York, 1966.
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