Non-Associative Division Algebra with Real Scalars has Dimension of Power of 2

Theorem

Let $A$ be a non-associative division algebra with real scalars.

Then the dimension of $A$ is a power of $2$.

Proof

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Historical Note

Non-Associative Division Algebra with Real Scalars has Dimension of Power of 2 was demonstrated by Heinz Hopf in $1940$.

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