Editing Field (mathematics) (section)

== Examples ==

=== Rational numbers ===
{{Main|Rational number}}
Rational numbers have been widely used a long time before the elaboration of the concept of field.
They are numbers that can be written as [[fraction (mathematics)|fractions]]
{{math|''a''/''b''}}, where {{math|''a''}} and {{math|''b''}} are [[integer]]s, and {{math|''b'' ≠ 0}}. The additive inverse of such a fraction is {{math|−''a''/''b''}}, and the multiplicative inverse (provided that {{math|''a'' ≠ 0}}) is {{math|''b''/''a''}}, which can be seen as follows:
: <math> \frac b a \cdot \frac a b = \frac{ba}{ab} = 1.</math>

The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows:<ref>{{harvp|Beachy|Blair|2006|loc=p. 120, Ch. 3}}</ref>
: <math>
\begin{align}
& \frac a b \cdot \left(\frac c d + \frac e f \right) \\[6pt]
= {} & \frac a b  \cdot \left(\frac c d  \cdot \frac f f + \frac e f \cdot \frac d d \right) \\[6pt]
= {} & \frac{a}{b} \cdot \left(\frac{cf}{df} + \frac{ed}{fd}\right) = \frac{a}{b} \cdot \frac{cf + ed}{df} \\[6pt]
= {} & \frac{a(cf + ed)}{bdf} = \frac{acf}{bdf} +  \frac{aed}{bdf} = \frac{ac}{bd} +  \frac{ae}{bf} \\[6pt]
= {} & \frac a b \cdot \frac c d + \frac a b \cdot \frac e f.
\end{align}
</math>

=== Real and complex numbers ===
[[File:Complex multi.svg|thumb|255px|The multiplication of complex numbers can be visualized geometrically by rotations and scalings.]]
{{main|Real number| Complex number}}

The [[real number]]s {{math|'''R'''}}, with the usual operations of addition and multiplication, also form a field. The [[complex number]]s {{math|'''C'''}} consist of expressions
: {{math|''a'' + ''bi'',}} with {{math|''a'', ''b''}} real,
where {{math|''i''}} is the [[imaginary unit]], i.e., a (non-real) number satisfying {{math|1=''i''<sup>2</sup> = −1}}.
Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for {{math|'''C'''}}. For example, the distributive law enforces
: {{math|1=(''a'' + ''bi'')(''c'' + ''di'') = ''ac'' + ''bci'' + ''adi'' + ''bdi''<sup>2</sup> = (''ac'' − ''bd'') + (''bc'' + ''ad'')''i''.}}
It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the [[Plane (geometry)|plane]], with [[Cartesian coordinates]] given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.

=== Constructible numbers ===
[[File:Root_construction_geometric_mean5.svg|thumb|255px|The [[geometric mean theorem]] asserts that {{math|1=''h''<sup>2</sup> = ''pq''}}. Choosing {{math|1=''q'' = 1}} allows construction of the square root of a given constructible number {{math|''p''}}.]]
{{main|Constructible numbers}}

In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with [[compass and straightedge]]. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of [[constructible numbers]].<ref>{{harvp|Artin|1991|loc=Chapter 13.4}}</ref> Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only [[Compass (drawing tool)|compass]] and [[straightedge]]. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field {{math|'''Q'''}} of rational numbers. The illustration shows the construction of [[square root]]s of constructible numbers, not necessarily contained within {{math|'''Q'''}}. Using the labeling in the illustration, construct the segments {{math|''AB''}}, {{math|''BD''}}, and a [[semicircle]] over {{math|''AD''}} (center at the [[midpoint]] {{math|''C''}}), which intersects the [[perpendicular]] line through {{math|''B''}} in a point {{math|''F''}}, at a distance of exactly <math>h=\sqrt p</math> from {{math|''B''}} when {{math|''BD''}} has length one.

Not all real numbers are constructible. It can be shown that <math>\sqrt[3] 2</math> is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a [[Doubling the cube|cube with volume 2]], another problem posed by the ancient Greeks.

=== 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)}}.