Frobenius theorem (real division algebras).html |
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In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite dimensional associative division algebras over the real numbers. The theorem proves that the only associative division algebra which is not commutative over the real numbers is the quaternions.
If D is a finite dimensional division algebra over the real numbers R then one of the following cases holds
- D = R
- D = C (complex numbers)
- D isomorphic to H (quaternions).
Pontryagin variant
If D is a connected, locally compact division ring, then either D = R, or D = C, or D = H.
References
- Ferdinand Georg Frobenius (1878) "Ueber lineare Substitutionen und bilineare Formen", Journal für die reine und angewandte Mathematik 84:1-63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp.343-405.
- Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp.30-2 ISBN 0-7923-2459-5.
- R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366-8.
- Lev Semenovich Pontryagin, Topological Groups, page 159, 1966.