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