Editing Field (mathematics) (section)

=== A field with four elements ===
{{main|Finite field#Field with four elements}}
{|class="wikitable floatright"
|+
! scope="col" style="float:text-align:center;"| Addition
! scope="col" style="float:text-align:center'"| Multiplication
|-
! scope="row" |
{| class="wikitable"
|-
! style="width:20%;"| + !! style="width:20%;"| {{math|''O''}} !! style="width:20%;"| {{math|''I''}} !! style="width:20%;"| {{math|''A''}} !! style="width:20%;"| {{math|''B''}}
|-
! {{math|''O''}}
| style="background:#fdd;"| {{color|blue| {{math|''O''}}}}
| style="background:#fdd;"| {{color|blue| {{math|''I''}}}}
|| {{math|''A''}}
|| {{math|''B''}}
|-
! {{math|''I''}}
| style="background:#fdd;"| {{color|blue| {{math|''I''}}}}
| style="background:#fdd;"| {{color|blue| {{math|''O''}}}}
|| {{math|''B''}}
|| {{math|''A''}}
|-
! {{math|''A''}}
|| {{math|''A''}}
|| {{math|''B''}}
|| {{math|''O''}}
|| {{math|''I''}}
|-
! {{math|''B''}}
|| {{math|''B''}}
|| {{math|''A''}}
|| {{math|''I''}}
|| {{math|''O''}}
|}
! scope="row" |
{| class="wikitable"

|-
! style="width:20%;"| ⋅ !! style="width:20%;"| {{math|''O''}} !! style="width:20%;"| {{math|''I''}} !! style="width:20%;"| {{math|''A''}} !! style="width:20%;"| {{math|''B''}}
|-
! {{math|''O''}}
| style="background:#fdd;"|{{color|blue| {{math|''O''}}}}
| style="background:#fdd;"|{{color|blue| {{math|''O''}}}}
|| {{math|''O''}}
|| {{math|''O''}}
|-
! {{math|''I''}}
| style="background:#fdd;"|{{color|blue| {{math|''O''}}}}
| style="background:#fdd;"|{{color|blue| {{math|''I''}}}}
|| {{math|''A''}}
|| {{math|''B''}}
|-
! {{math|''A''}}
|| {{math|''O''}}
|| {{math|''A''}}
|| {{math|''B''}}
|| {{math|''I''}}
|-
! {{math|''B''}}
|| {{math|''O''}}
|| {{math|''B''}}
|| {{math|''I''}}
|| {{math|''A''}}
|}
|}

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called {{math|''O''}}, {{math|''I''}}, {{math|''A''}}, and {{math|''B''}}. The notation is chosen such that {{math|''O''}} plays the role of the additive identity element (denoted 0 in the axioms above), and {{math|''I''}} is the multiplicative identity (denoted {{math|1}} in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example,
: {{math|1=''A'' ⋅ (''B'' + ''A'') = ''A'' ⋅ ''I'' = ''A''}}, which equals {{nowrap|1={{math|1=''A'' ⋅ ''B'' + ''A'' ⋅ ''A'' = ''I'' + ''B'' = ''A''}}}}, as required by the distributivity.

This field is called a [[finite field]] or '''Galois field''' with four elements, and is denoted {{math|'''F'''<sub>4</sub>}} or {{math|GF(4)}}.<ref>{{harvp|Lidl|Niederreiter|2008|loc=Example 1.62}}</ref> The [[subset]] consisting of {{math|''O''}} and {{math|''I''}} (highlighted in red in the tables at the right) is also a field, known as the ''[[binary field]]'' {{math|'''F'''<sub>2</sub>}} or {{math|GF(2)}}.