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. 1.

$k$,

2. 2.

$\left(\frac{\alpha}{k}\right)$ for $\alpha\in k$,

3. 3.

$\left(\frac{\alpha,\beta}{k}\right)$ for $\alpha,\beta\in k$,

4. 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 and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.
Title Hurwitz’s theorem on composition algebras HurwitzsTheoremOnCompositionAlgebras 2013-03-22 17:18:20 2013-03-22 17:18:20 Algeboy (12884) Algeboy (12884) 4 Algeboy (12884) Theorem msc 17A75 JacobsonsTheoremOnCompositionAlgebras