Editing Field (mathematics) (section)

=== 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}}.