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

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
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