Editing Field (mathematics) (section)

=== Division rings ===
Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of a ''[[division ring]]'' or ''skew field'';{{efn|Historically, division rings were sometimes referred to as fields, while fields were called ''commutative fields''.}} sometimes associativity is weakened as well. The only division rings that are finite-dimensional {{math|'''R'''}}-vector spaces are {{math|'''R'''}} itself, {{math|'''C'''}} (which is a field), and the [[quaternion]]s {{math|'''H'''}} (in which multiplication is non-commutative). This result is known as the [[Frobenius_theorem_(real_division_algebras)|Frobenius theorem]]. The [[octonion]]s {{math|'''O'''}}, for which multiplication is neither commutative nor associative, is a normed [[Alternative_algebra|alternative]] division algebra, but is not a division ring. This fact was proved using methods of [[algebraic topology]] in 1958 by [[Michel Kervaire]], [[Raoul Bott]], and [[John Milnor]].<ref>{{harvp|Baez|2002}}</ref> 

[[Wedderburn's little theorem]] states that all finite [[Division ring|division rings]] are fields.