# 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

## 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