Editing Field (mathematics) (section)

=== 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=§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|−1}}, since {{math|1=0 = 1 + (−1)}} and {{math|1=−''a'' = (−1)''a''}}.{{efn|The a priori twofold use of the symbol "{{math|−}}" for denoting one part of a constant and for the additive inverses is justified by this latter condition.}}