Frobenius theorem (real division algebras): Difference between revisions - Wikipedia

Line 57:

==Remarks and related results==

*The fact that {{mvar|D}} is generated by {{math|''e''₁, ..., ''e_n''}} subject to the above relations means that {{mvar|D}} is the [[Clifford algebra]] of {{math|'''R'''^''n''}}. The last step shows that the only real Clifford algebras which are division algebras are {{math|Cℓ⁰, Cℓ¹}} and {{math|Cℓ²}}.

*As a consequence, the only [[commutative]] division algebras are {{math|'''R'''}} and {{math|'''C'''}}. Also note that {{math|'''H'''}} is not a {{math|'''C'''}}-algebra. If it were, then the center of {{math|'''H'''}} has to contain {{math|'''C'''}}, but the center of {{math|'''H'''}} is {{math|'''R'''}}. Therefore, the only finite-dimensional division algebra over {{math|'''C'''}} is {{math|'''C'''}} itself.

* This theorem is closely related to [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]], which states that the only real [[normed division algebra]]s are {{math|'''R''', '''C''', '''H'''}}, and the (non-associative) algebra [[octonions|{{math|'''O'''}}]].

* '''Pontryagin variant.''' If {{mvar|D}} is a [[connected space|connected]], [[locally compact space|locally compact]] division [[topological ring|ring]], then {{math|''D'' {{=}} '''R''', '''C'''}}, or {{math|'''H'''}}.