Bott-Milnor-Kervaire 1,2,4,8 Theorem

Theorem

Let $A$ be a division algebra with real scalars.

Then the dimension of $A$ is either:

$1$: the real numbers $\R$

$2$: the complex numbers $\C$

$4$: the quaternions $\Bbb H$

or:

$8$: the octonions $\Bbb O$.

Proof

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Source of Name

This entry was named for Raoul Bott, John Willard Milnor and Michel André Kervaire.

Historical Note

Bott-Milnor-Kervaire 1,2,4,8 Theorem was proved by Raoul Bott, John Willard Milnor and Michel André Kervaire in $1958$.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem