Editing Field (mathematics) (section)

== Related notions ==
In addition to the additional structure that fields may enjoy, fields admit various other related notions. Since in any field {{math|0 ≠ 1}}, any field has at least two elements. Nonetheless, there is a concept of [[field with one element]], which is suggested to be a limit of the finite fields {{math|'''F'''<sub>''p''</sub>}}, as {{math|''p''}} tends to {{math|1}}.<ref>{{harvp|Tits|1957}}</ref> In addition to division rings, there are various other weaker algebraic structures related to fields such as [[quasifield]]s, [[Near-field (mathematics)|near-field]]s and [[semifield]]s.

There are also [[proper class]]es with field structure, which are sometimes called '''Field'''s, with a capital 'F'. The [[surreal number]]s form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The [[nimber]]s, a concept from [[game theory]], form such a Field as well.<ref>{{harvp|Conway|1976}}</ref>

=== 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.