Editing Field (mathematics) (section)

=== Classic definition ===
Formally, a field is a [[set (mathematics)|set]] {{math|''F''}} together with two [[binary operation]]s on {{mvar|F}} called ''addition'' and ''multiplication''.<ref>{{harvp|Beachy|Blair|2006|loc=Definition 4.1.1, p.&nbsp;181}}</ref> A binary operation on {{mvar|F}} is a mapping {{math|''F'' × ''F'' → ''F''}}, that is, a correspondence that associates with each ordered pair of elements of {{mvar|F}} a uniquely determined element of {{mvar|F}}.<ref>{{harvp|Fraleigh|1976|p=10}}</ref><ref>{{harvp|McCoy|1968|p=16}}</ref> The result of the addition of {{math|''a''}} and {{math|''b''}} is called the sum of {{math|''a''}} and {{math|''b''}}, and is denoted {{math|''a'' + ''b''}}. Similarly, the result of the multiplication of {{math|''a''}} and {{math|''b''}} is called the product of {{math|''a''}} and {{math|''b''}}, and is denoted {{math|''a'' ⋅ ''b''}}. These operations are required to satisfy the following properties, referred to as ''[[Axiom#Non-logical axioms|field axioms]]''. 

These axioms are required to hold for all [[element (mathematics)|element]]s  {{mvar|a}}, {{mvar|b}}, {{mvar|c}} of the field {{mvar|F}}:
* [[Associativity]] of addition and multiplication: {{math|1=''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c''}}, and {{math|1=''a'' ⋅ (''b'' ⋅ ''c'') = (''a'' ⋅ ''b'') ⋅ ''c''}}.
* [[Commutativity]] of addition and multiplication: {{math|1=''a'' + ''b'' = ''b'' + ''a''}}, and {{math|1=''a'' ⋅ ''b'' = ''b'' ⋅ ''a''}}.
* [[Additive identity|Additive]] and [[multiplicative identity]]: there exist two distinct elements {{math|0}} and {{math|1}} in {{math|''F''}} such that {{math|1=''a'' + 0 = ''a''}} and {{math|1=''a'' ⋅ 1 = ''a''}}.
* [[Additive inverse]]s: for every {{math|''a''}} in {{math|''F''}}, there exists an element in {{math|''F''}}, denoted {{math|−''a''}}, called the ''additive inverse'' of {{math|''a''}}, such that {{math|1=''a'' + (−''a'') = 0}}.
* [[Multiplicative inverse]]s: for every {{math|''a'' ≠ 0}} in {{math|''F''}}, there exists an element in {{math|''F''}}, denoted by {{math|''a''<sup>−1</sup>}} or {{math|1/''a''}}, called the ''multiplicative inverse'' of {{math|''a''}}, such that {{math|1=''a'' ⋅ ''a''<sup>−1</sup> = 1}}.
* [[Distributivity]] of multiplication over addition: {{math|1=''a'' ⋅ (''b'' + ''c'') = (''a'' ⋅ ''b'') + (''a'' ⋅ ''c'')}}.

An equivalent, and more succinct, definition is: a field has two commutative operations, called addition and multiplication; it is a [[group (mathematics)|group]] under addition with {{math|0}} as the additive identity; the nonzero elements form a group under multiplication with {{math|1}} as the multiplicative identity; and multiplication distributes over addition.

Even more succinctly: a field is a [[commutative ring]] where {{math|0 ≠ 1}} and all nonzero elements are [[unit (ring theory)|invertible]] under multiplication.