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== Definition == Informally, a field is a set, along with two [[binary operation|operation]]s defined on that set: an addition operation {{math|''a'' + ''b''}} and a multiplication operation {{math|''a'' β ''b''}}, both of which behave similarly as they do for [[rational number]]s and [[real number]]s. This includes the existence of an [[additive inverse]] {{math|β''a''}} for all elements {{mvar|a}} and of a [[multiplicative inverse]] {{math|''b''<sup>β1</sup>}} for every nonzero element {{mvar|b}}. This allows the definition of the so-called ''inverse operations'', subtraction {{math|''a'' β ''b''}} and division {{math|''a'' / ''b''}}, as {{math|1=''a'' β ''b'' = ''a'' + (β''b'')}} and {{math|1=''a'' / ''b'' = ''a'' β ''b''<sup>β1</sup>}}. Often the product {{math|''a'' β ''b''}} is represented by juxtaposition, as {{mvar|ab}}. === 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. 181}}</ref> A binary operation on {{mvar|F}} is a mapping {{math|''F'' Γ ''F'' β ''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'' β ''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'' β (''b'' β ''c'') = (''a'' β ''b'') β ''c''}}. * [[Commutativity]] of addition and multiplication: {{math|1=''a'' + ''b'' = ''b'' + ''a''}}, and {{math|1=''a'' β ''b'' = ''b'' β ''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'' β 1 = ''a''}}. * [[Additive inverse]]s: for every {{math|''a''}} in {{math|''F''}}, there exists an element in {{math|''F''}}, denoted {{math|β''a''}}, called the ''additive inverse'' of {{math|''a''}}, such that {{math|1=''a'' + (β''a'') = 0}}. * [[Multiplicative inverse]]s: for every {{math|''a'' β 0}} in {{math|''F''}}, there exists an element in {{math|''F''}}, denoted by {{math|''a''<sup>β1</sup>}} or {{math|1/''a''}}, called the ''multiplicative inverse'' of {{math|''a''}}, such that {{math|1=''a'' β ''a''<sup>β1</sup> = 1}}. * [[Distributivity]] of multiplication over addition: {{math|1=''a'' β (''b'' + ''c'') = (''a'' β ''b'') + (''a'' β ''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 β 1}} and all nonzero elements are [[unit (ring theory)|invertible]] under multiplication. === Alternative definition === Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties. [[Division by zero]] is, by definition, excluded.<ref>{{harvp|Clark|1984|loc=Chapter 3}}</ref> In order to avoid [[existential quantifier]]s, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two [[arity#Nullary|nullary]] operations (the constants {{math|0}} and {{math|1}}). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in [[constructive mathematics]] and [[computing]].<ref>{{harvp|Mines|Richman|Ruitenburg|1988|loc=Β§II.2}}. See also [[Heyting field]].</ref> One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two (not necessarily distinct) constants {{math|1}} and {{math|β1}}, since {{math|1=0 = 1 + (β1)}} and {{math|1=β''a'' = (β1)''a''}}.{{efn|The a priori twofold use of the symbol "{{math|β}}" for denoting one part of a constant and for the additive inverses is justified by this latter condition.}}
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