Editing Field (mathematics) (section)

== Elementary notions ==
In this section, {{math|''F''}} denotes an arbitrary field and {{math|''a''}} and {{math|''b''}} are arbitrary [[element (set theory)|elements]] of {{math|''F''}}.

=== Consequences of the definition ===
One has {{math|1=''a'' ⋅ 0 = 0}} and {{math|1=−''a'' = (−1) ⋅ ''a''}}. In particular, one may deduce the additive inverse of every element as soon as one knows {{math|−1}}.<ref>{{harvp|Beachy|Blair|2006|loc=p. 120, Ch. 3}}</ref>

If {{math|1=''ab'' = 0}} then {{math|1=''a''}} or {{math|''b''}} must be {{math|0}}, since, if {{math|''a'' ≠ 0}}, then
{{math|1=''b'' = (''a''<sup>−1</sup>''a'')''b'' = ''a''<sup>−1</sup>(''ab'')  = ''a''<sup>−1</sup> ⋅ 0 = 0}}. This means that every field is an [[integral domain]].

In addition, the following properties are true for any elements {{math|''a''}} and {{math|''b''}}:
: {{math|1=−0 = 0}}
: {{math|1=1<sup>−1</sup> = 1}}
: {{math|1=(−(−''a'')) = ''a''}}
: {{math|1=(−''a'') ⋅ ''b'' = ''a'' ⋅ (−''b'') = −(''a'' ⋅ ''b'')}}
: {{math|1=(''a''<sup>−1</sup>)<sup>−1</sup> = ''a''}} if {{math|''a'' ≠ 0}}

=== 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}}'').

=== Characteristic ===
In addition to the multiplication of two elements of {{math|''F''}}, it is possible to define the product {{math|''n'' ⋅ ''a''}} of an arbitrary element {{math|''a''}} of {{math|''F''}} by a positive [[integer]] {{math|''n''}} to be the {{math|''n''}}-fold sum
: {{math|''a'' + ''a'' + ... + ''a''}} (which is an element of {{math|''F''}}.)

If there is no positive integer such that
: {{math|1=''n'' ⋅ 1 = 0}},
then {{math|''F''}} is said to have [[characteristic (algebra)|characteristic]] {{math|0}}.<ref>{{harvp|Adamson|2007|loc=§I.2, p.&nbsp;10}}</ref> For example, the field of rational numbers {{math|'''Q'''}} has characteristic 0 since no positive integer {{math|''n''}} is zero. Otherwise, if there ''is'' a positive integer {{math|''n''}} satisfying this equation, the smallest such positive integer can be shown to be a [[prime number]]. It is usually denoted by {{math|''p''}} and the field is said to have characteristic {{math|''p''}} then.
For example, the field {{math|'''F'''<sub>4</sub>}} has characteristic {{math|2}} since (in the notation of the above addition table) {{math|1=''I'' + ''I'' = O }}.

If {{math|''F''}} has characteristic {{math|''p''}}, then {{math|1=''p'' ⋅ ''a'' = 0}} for all {{math|''a''}} in {{math|''F''}}. This implies that
: {{math|1=(''a'' + ''b'')<sup>''p''</sup> = {{itco|''a''}}<sup>''p''</sup> + {{itco|''b''}}<sup>''p''</sup>}},
since all other [[binomial coefficient]]s appearing in the [[binomial formula]] are divisible by {{math|''p''}}. Here, {{math|1={{itco|''a''}}<sup>''p''</sup> := ''a'' ⋅ ''a'' ⋅ ⋯ ⋅ ''a''}} ({{math|''p''}} factors) is the {{math|''p''}}th power, i.e., the {{math|''p''}}-fold product of the element {{math|''a''}}. Therefore, the [[Frobenius map]]
: {{math|''F'' → ''F'' : ''x'' ↦ {{itco|''x''}}<sup>''p''</sup>}}
is compatible with the addition in {{math|''F''}} (and also with the multiplication), and is therefore a field homomorphism.<ref>{{harvp|Escofier|2012|loc=14.4.2}}</ref> The existence of this homomorphism makes fields in characteristic {{math|''p''}} quite different from fields of characteristic {{math|0}}.

=== Subfields and prime fields<span class="anchor" id="Prime field"></span> ===
A ''[[field extension|subfield]]'' {{math|''E''}} of a field {{math|''F''}} is a subset of {{math|''F''}} that is a field with respect to the field operations of {{math|''F''}}. Equivalently {{math|''E''}} is a subset of {{math|''F''}} that contains {{math|1}}, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that {{math|1 ∊ ''E''}}, that for all {{math|''a'', ''b'' ∊ ''E''}} both {{math|''a'' + ''b''}} and {{math|''a'' ⋅ ''b''}} are in {{math|''E''}}, and that for all {{math|''a'' ≠ 0}} in {{math|''E''}}, both {{math|−''a''}} and {{math|1/''a''}} are in {{math|''E''}}.

[[Field homomorphism]]s are maps {{math|''φ'': ''E'' → ''F''}} between two fields such that {{math|1=''φ''(''e''<sub>1</sub> + ''e''<sub>2</sub>) = ''φ''(''e''<sub>1</sub>) + ''φ''(''e''<sub>2</sub>)}}, {{math|1=''φ''(''e''<sub>1</sub>''e''<sub>2</sub>) = ''φ''(''e''<sub>1</sub>)&hairsp;''φ''(''e''<sub>2</sub>)}}, and {{math|1=''φ''(1<sub>''E''</sub>) = 1<sub>''F''</sub>}}, where {{math|''e''<sub>1</sub>}} and {{math|''e''<sub>2</sub>}} are arbitrary elements of {{math|''E''}}. All field homomorphisms are [[injective]].<ref>{{harvp|Adamson|2007|loc=§I.3}}</ref> If {{math|''φ''}} is also [[surjective]], it is called an [[isomorphism]] (or the fields {{math|''E''}} and {{math|''F''}} are called isomorphic).

A field is called a '''prime field''' if it has no proper (i.e., strictly smaller) subfields. Any field {{math|''F''}} contains a prime field. If the [[Characteristic (algebra)|characteristic]] of {{math|''F''}} is {{math|''p''}} (a prime number), the prime field is isomorphic to the finite field {{math|'''F'''<sub>''p''</sub>}} introduced below. Otherwise the prime field is isomorphic to {{math|'''Q'''}}.<ref>{{harvp|Adamson|2007|loc=p. 12}}</ref>