Non-Commutative Finite-Dimensional Associative Division Algebra over Real Numbers is Set of Quaternions

Theorem

Up to isomorphism, the quaternions form the only non-commutative, finite-dimensional associative division algebra over the set of real numbers $\R$.

Proof

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Also see

• Frobenius's Theorem

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): linear algebra: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Frobenius's theorem