Editing Field (mathematics) (section)

=== Additive and multiplicative groups of a field ===
The axioms of a field {{math|''F''}} imply that it is an [[abelian group]] under addition. This group is called the [[additive group]] of the field, and is sometimes denoted by {{math|(''F'', +)}} when denoting it simply as {{math|''F''}} could be confusing.

Similarly, the ''nonzero'' elements of {{math|''F''}} form an abelian group under multiplication, called the [[multiplicative group]], and denoted by <math>(F \smallsetminus \{0\}, \cdot)</math> or just <math>F \smallsetminus \{0\}</math>, or {{math|''F''<sup>×</sup>}}.

A field may thus be defined as set {{math|''F''}} equipped with two operations denoted as an addition and a multiplication such that {{math|''F''}} is an abelian group under addition, <math>F \smallsetminus \{0\}</math> is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is [[distributive property|distributive]] over addition.{{efn|Equivalently, a field is an [[algebraic structure]] {{math|⟨''F'', +, ⋅, −, <sup>−1</sup>, 0, 1⟩}} of type {{math|{{angle bracket|2, 2, 1, 1, 0, 0}}}}, such that {{math|0<sup>−1</sup>}} is not defined, {{math|{{angle bracket|''F'', +, −, 0}}}} and <math>\left\langle F \smallsetminus \{0\}, \cdot, {}^{-1}\right\rangle</math> are abelian groups, and {{math|⋅}} is distributive over {{math|+}}.<ref>{{harvp|Wallace|1998|loc=Th. 2}}</ref>}} Some elementary statements about fields can therefore be obtained by applying general facts of [[group (mathematics)|groups]]. For example, the additive and multiplicative inverses {{math|−''a''}} and {{math|''a''<sup>−1</sup>}} are uniquely determined by {{math|''a''}}.

The requirement {{math|1 ≠ 0}} is imposed by convention to exclude the [[trivial ring]], which consists of a single element; this guides any choice of the axioms that define fields.

Every finite [[subgroup]] of the multiplicative group of a field is [[cyclic group|cyclic]] (see ''{{slink|Root of unity|Cyclic groups}}'').