Editing Field (mathematics) (section)

== Finite fields ==
{{Main|Finite field}}
''Finite fields'' (also called ''Galois fields'') are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example {{math|'''F'''<sub>4</sub>}} is a field with four elements. Its subfield {{math|'''F'''<sub>2</sub>}} is the smallest field, because by definition a field has at least two distinct elements, {{math|0}} and {{math|1}}.

[[File:Clock group.svg|thumb|In modular arithmetic modulo&nbsp;12, {{math|1=9 + 4 = 1}} since {{math|1=9 + 4 = 13}} in {{math|'''Z'''}}, which divided by {{math|12}} leaves remainder&nbsp;{{math|1}}. However, {{math|'''Z'''/12'''Z'''}} is not a field because {{math|12}} is not a prime number.]]
The simplest finite fields, with prime order, are most directly accessible using [[modular arithmetic]]. For a fixed positive integer {{math|''n''}}, arithmetic "modulo {{math|''n''}}" means to work with the numbers
: {{math|1='''Z'''/''n'''''Z''' = {0, 1, ..., ''n'' − 1}.}}
The addition and multiplication on this set are done by performing the operation in question in the set {{math|'''Z'''}} of integers, dividing by {{math|''n''}} and taking the remainder as result. This construction yields a field precisely if {{math|''n''}} is a [[prime number]]. For example, taking the prime {{math|1=''n'' = 2}} results in the above-mentioned field {{math|'''F'''<sub>2</sub>}}. For {{math|1=''n'' = 4}} and more generally, for any [[composite number]] (i.e., any number {{math|''n''}} which can be expressed as a product {{math|1=''n'' = ''r'' ⋅ ''s''}} of two strictly smaller natural numbers), {{math|1='''Z'''/''n'''''Z'''}} is not a field: the product of two non-zero elements is zero since {{math|1=''r'' ⋅ ''s'' = 0}} in {{math|'''Z'''/''n'''''Z'''}}, which, as was explained [[#Consequences of the definition|above]], prevents {{math|'''Z'''/''n'''''Z'''}} from being a field. The field {{math|'''Z'''/''p'''''Z'''}} with {{math|''p''}} elements ({{math|''p''}} being prime) constructed in this way is usually denoted by {{math|'''F'''<sub>''p''</sub>}}.

Every finite field {{math|''F''}} has {{math|1=''q'' = ''p''<sup>''n''</sup>}} elements, where {{math|1=''p''}} is prime and {{math|''n'' ≥ 1}}. This statement holds since {{math|''F''}} may be viewed as a [[vector space]] over its prime field. The [[dimension of a vector space|dimension]] of this vector space is necessarily finite, say {{math|''n''}}, which implies the asserted statement.<ref>{{harvp|Lidl|Niederreiter|2008|loc=Lemma 2.1, Theorem 2.2}}</ref>

A field with {{math|1=''q'' = ''p''<sup>''n''</sup>}} elements can be constructed as the [[splitting field]] of the [[polynomial]]
: {{math|1={{itco|''f''}}(''x'') = {{itco|''x''}}<sup>''q''</sup> − ''x''}}.
Such a splitting field is an extension of {{math|'''F'''<sub>''p''</sub>}} in which the polynomial {{math|''f''}} has {{math|''q''}} zeros. This means {{math|''f''}} has as many zeros as possible since the [[degree of a polynomial|degree]] of {{math|''f''}} is {{math|''q''}}. For {{math|1=''q'' = 2<sup>2</sup> = 4}}, it can be checked case by case using the above multiplication table that all four elements of {{math|'''F'''<sub>4</sub>}} satisfy the equation {{math|1=''x''<sup>4</sup> = ''x''}}, so they are zeros of {{math|''f''}}. By contrast, in {{math|'''F'''<sub>2</sub>}}, {{math|''f''}} has only two zeros (namely {{math|0}} and {{math|1}}), so {{math|''f''}} does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic.<ref>{{harvp|Lidl|Niederreiter|2008|loc=Theorem 1.2.5}}</ref> It is thus customary to speak of ''the'' finite field with {{math|''q''}} elements, denoted by {{math|'''F'''<sub>''q''</sub>}} or {{math|GF(''q'')}}.