The only finite-dimensional associative division algebras over the real numbers are the real numbers themselves, the complex numbers, and the quaternions. Proved by Ferdinand Georg Frobenius, it explains why no larger associative number system of that kind exists.
Facts
StatementEvery finite-dimensional associative division algebra over the real numbers is isomorphic to the real numbers, the complex numbers, or the quaternions. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
1. Frobenius theorem (real division algebras) (Wikipedia)
Wikimedia FoundationLead section, isomorphism classes sentence
According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers), C (the complex numbers), or H (the quaternions).
Lead section, first sentence
the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers.
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