Editing Field (mathematics) (section)

== Constructing fields ==

=== Constructing fields from rings ===
A [[commutative ring]] is a set that is equipped with an addition and multiplication operation and satisfies all the axioms of a field, except for the existence of multiplicative inverses {{math|''a''<sup>−1</sup>}}.<ref>{{harvp|Lang|2002|loc=§II.1}}</ref> For example, the integers {{math|'''Z'''}} form a commutative ring, but not a field: the [[Multiplicative inverse|reciprocal]] of an integer {{math|''n''}} is not itself an integer, unless {{math|1=''n'' = ±1}}.

In the hierarchy of algebraic structures fields can be characterized as the commutative rings {{math|''R''}} in which every nonzero element is a [[unit (ring theory)|unit]] (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct [[Ideal (ring theory)|ideal]]s, {{math|(0)}} and {{math|''R''}}. Fields are also precisely the commutative rings in which {{math|(0)}} is the only [[prime ideal]].

Given a commutative ring {{math|''R''}}, there are two ways to construct a field related to {{math|''R''}}, i.e., two ways of modifying {{math|''R''}} such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of {{math|'''Z'''}} is {{math|'''Q'''}}, the rationals, while the residue fields of {{math|'''Z'''}} are the finite fields {{math|'''F'''<sub>''p''</sub>}}.

==== Field of fractions ====
Given an [[integral domain]] {{math|''R''}}, its [[field of fractions]] {{math|''Q''(''R'')}} is built with the fractions of two elements of {{math|''R''}} exactly as '''Q''' is constructed from the integers. More precisely, the elements of {{math|''Q''(''R'')}} are the fractions {{math|''a''/''b''}} where {{math|''a''}} and {{math|''b''}} are in {{math|''R''}}, and {{math|''b'' ≠ 0}}. Two fractions {{math|''a''/''b''}} and {{math|''c''/''d''}} are equal if and only if {{math|1=''ad'' = ''bc''}}. The operation on the fractions work exactly as for rational numbers. For example,
: <math>\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd}.</math>
It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field.<ref>{{harvp|Artin|1991|loc=§10.6}}</ref>

The field {{math|''F''(''x'')}} of the [[rational fraction]]s over a field (or an integral domain) {{math|''F''}} is the field of fractions of the [[polynomial ring]] {{math|''F''[''x'']}}. The field {{math|''F''((''x''))}} of [[Laurent series]]
: <math>\sum_{i=k}^\infty a_i x^i \ (k \in \Z, a_i \in F)</math>
over a field {{math|''F''}} is the field of fractions of the ring {{math|''F''<nowiki>[[</nowiki>''x'']]}} of [[formal power series]] (in which {{math|''k'' ≥ 0}}). Since any Laurent series is a fraction of a power series divided by a power of {{math|''x''}} (as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.

==== Residue fields ====
In addition to the field of fractions, which embeds {{math|''R''}} [[injective map|injectively]] into a field, a field can be obtained from a commutative ring {{math|''R''}} by means of a [[surjective map]] onto a field {{math|''F''}}. Any field obtained in this way is a [[quotient ring|quotient]] {{math|{{nowrap|''R'' / ''m''}}}}, where {{math|''m''}} is a [[maximal ideal]] of {{math|''R''}}. If {{math|''R''}} [[local ring|has only one maximal ideal]] {{math|''m''}}, this field is called the [[residue field]] of {{math|''R''}}.<ref>{{harvp|Eisenbud|1995|loc=p. 60}}</ref>

The [[principal ideal|ideal generated by a single polynomial]] {{math|''f''}} in the polynomial ring {{math|1=''R'' = ''E''[''X'']}} (over a field {{math|''E''}}) is maximal if and only if {{math|''f''}} is [[irreducible polynomial|irreducible]] in {{math|''E''}}, i.e., if {{math|''f''}} cannot be expressed as the product of two polynomials in {{math|''E''[''X'']}} of smaller [[degree of a polynomial|degree]]. This yields a field
: {{math|1=''F'' = ''E''[''X''] / ({{itco|''f''}}(''X'')).}}
This field {{math|''F''}} contains an element {{math|''x''}} (namely the [[residue class]] of {{math|''X''}}) which satisfies the equation
: {{math|1={{itco|''f''}}(''x'') = 0}}.
For example, {{math|'''C'''}} is obtained from {{math|'''R'''}} by [[adjunction (field theory)|adjoining]] the [[imaginary unit]] symbol {{mvar|i}}, which satisfies {{math|1={{itco|''f''}}(''i'') = 0}}, where {{math|1={{itco|''f''}}(''X'') = ''X''<sup>2</sup> + 1}}. Moreover, {{math|''f''}} is irreducible over {{math|'''R'''}}, which implies that the map that sends a polynomial {{math|{{itco|''f''}}(''X'') ∊ '''R'''[''X'']}} to {{math|{{itco|''f''}}(''i'')}} yields an isomorphism
: <math>\mathbf R[X]/\left(X^2 + 1\right) \ \stackrel \cong \longrightarrow \ \mathbf C.</math>

=== Constructing fields within a bigger field ===
Fields can be constructed inside a given bigger container field. Suppose given a field {{math|''E''}}, and a field {{math|''F''}} containing {{math|''E''}} as a subfield. For any element {{math|''x''}} of {{math|''F''}}, there is a smallest subfield of {{math|''F''}} containing {{math|''E''}} and {{math|''x''}}, called the subfield of ''F'' generated by {{math|''x''}} and denoted {{math|''E''(''x'')}}.<ref>{{harvp|Jacobson|2009|loc=p. 213}}</ref> The passage from {{math|''E''}} to {{math|''E''(''x'')}} is referred to by ''[[adjunction (field theory)|adjoining]] an element'' to {{math|''E''}}. More generally, for a subset {{math|''S'' ⊂ ''F''}}, there is a minimal subfield of {{math|''F''}} containing {{math|''E''}} and {{math|''S''}}, denoted by {{math|''E''(''S'')}}.

The [[compositum]] of two subfields {{math|''E''}} and {{math|''E''{{′}}}} of some field {{math|''F''}} is the smallest subfield of {{math|''F''}} containing both {{math|''E''}} and {{math|''E''{{′}}}}. The compositum can be used to construct the biggest subfield of {{math|''F''}} satisfying a certain property, for example the biggest subfield of {{math|''F''}}, which is, in the language introduced below, algebraic over {{math|''E''}}.{{efn|Further examples include the maximal [[unramified extension]] or the maximal [[abelian extension]] within {{math|''F''}}.}}

=== Field extensions ===
{{See|Glossary of field theory}}
The notion of a subfield {{math|''E'' ⊂ ''F''}} can also be regarded from the opposite point of view, by referring to {{math|''F''}} being a ''[[field extension]]'' (or just extension) of {{math|''E''}}, denoted by
: {{math|''F'' / ''E''}},
and read "{{math|''F''}} over {{math|''E''}}".

A basic datum of a field extension is its [[degree of a field extension|degree]] {{math|[''F'' : ''E'']}}, i.e., the dimension of {{math|''F''}} as an {{math|''E''}}-vector space. It satisfies the formula<ref>{{harvp|Artin|1991|loc=Theorem 13.3.4}}</ref>
: {{math|1=[''G'' : ''E''] = [''G'' : ''F''] [''F'' : ''E'']}}.
Extensions whose degree is finite are referred to as finite extensions. The extensions {{math|'''C''' / '''R'''}} and {{math|'''F'''<sub>4</sub> / '''F'''<sub>2</sub>}} are of degree {{math|2}}, whereas {{math|'''R''' / '''Q'''}} is an infinite extension.

==== Algebraic extensions ====
A pivotal notion in the study of field extensions {{math|''F'' / ''E''}} are [[algebraic element]]s. An element {{math|''x'' ∈ ''F''}} is ''algebraic'' over {{mvar|E}} if it is a [[zero of a function|root]] of a [[polynomial]] with [[coefficient]]s in {{mvar|E}}, that is, if it satisfies a [[polynomial equation]]
: {{math|1=''e''<sub>''n''</sub>&hairsp;''x''<sup>''n''</sup> + ''e''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + ⋯ + ''e''<sub>1</sub>''x'' + ''e''<sub>0</sub> = 0}},
with {{math|''e''<sub>''n''</sub>, ..., ''e''<sub>0</sub>}} in {{mvar|E}}, and {{math|''e''<sub>''n''</sub> ≠ 0}}.
For example, the [[imaginary unit]] {{math|''i''}} in {{math|'''C'''}} is algebraic over {{math|'''R'''}}, and even over {{math|'''Q'''}}, since it satisfies the equation
: {{math|1=''i''<sup>2</sup> + 1 = 0}}.
A field extension in which every element of {{math|''F''}} is algebraic over {{math|''E''}} is called an [[algebraic extension]]. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.<ref>{{harvp|Artin|1991|loc=Corollary 13.3.6}}</ref>

The subfield {{math|''E''(''x'')}} generated by an element {{math|''x''}}, as above, is an algebraic extension of {{math|''E''}} if and only if {{math|''x''}} is an algebraic element. That is to say, if {{math|''x''}} is algebraic, all other elements of {{math|''E''(''x'')}} are necessarily algebraic as well. Moreover, the degree of the extension {{math|''E''(''x'') / ''E''}}, i.e., the dimension of {{math|''E''(''x'')}} as an {{math|''E''}}-vector space, equals the minimal degree {{math|''n''}} such that there is a polynomial equation involving {{math|''x''}}, as above. If this degree is {{math|''n''}}, then the elements of {{math|''E''(''x'')}} have the form
: <math>\sum_{k=0}^{n-1} a_k x^k, \ \ a_k \in E.</math>

For example, the field {{math|'''Q'''(''i'')}} of [[Gaussian rational]]s is the subfield of {{math|'''C'''}} consisting of all numbers of the form {{math|''a'' + ''bi''}} where both {{math|''a''}} and {{math|''b''}} are rational numbers: summands of the form {{math|''i''<sup>2</sup>}} (and similarly for higher exponents) do not have to be considered here, since {{math|''a'' + ''bi'' + ''ci''<sup>2</sup>}} can be simplified to {{math|''a'' − ''c'' + ''bi''}}.

==== Transcendence bases ====
The above-mentioned field of [[rational fraction]]s {{math|''E''(''X'')}}, where {{math|''X''}} is an [[indeterminate (variable)|indeterminate]], is not an algebraic extension of {{math|''E''}} since there is no polynomial equation with coefficients in {{math|''E''}} whose zero is {{math|''X''}}. Elements, such as {{math|''X''}}, which are not algebraic are called [[Algebraic element|transcendental]]. Informally speaking, the indeterminate {{math|''X''}} and its powers do not interact with elements of {{math|''E''}}. A similar construction can be carried out with a set of indeterminates, instead of just one.

Once again, the field extension {{math|''E''(''x'') / ''E''}} discussed above is a key example: if {{math|''x''}} is not algebraic (i.e., {{math|''x''}} is not a [[root of a function|root]] of a polynomial with coefficients in {{math|''E''}}), then {{math|''E''(''x'')}} is isomorphic to {{math|''E''(''X'')}}. This isomorphism is obtained by substituting {{math|''x''}} to {{math|''X''}} in rational fractions.

A subset {{math|''S''}} of a field {{math|''F''}} is a [[transcendence basis]] if it is [[algebraically independent]] (do not satisfy any polynomial relations) over {{math|''E''}} and if {{math|''F''}} is an algebraic extension of {{math|''E''(''S'')}}. Any field extension {{math|''F'' / ''E''}} has a transcendence basis.<ref>{{harvp|Bourbaki|1988|loc=Chapter V, §14, No. 2, Theorem 1}}</ref> Thus, field extensions can be split into ones of the form {{math|''E''(''S'') / ''E''}} ([[transcendental extension|purely transcendental extensions]]) and algebraic extensions.

=== Closure operations ===
A field is [[algebraically closed]] if it does not have any strictly bigger algebraic extensions or, equivalently, if any [[polynomial equation]]
: {{math|1=''f''<sub>''n''</sub>&hairsp;''x''<sup>''n''</sup> + ''f''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + ⋯ + ''f''<sub>1</sub>''x'' + ''f''<sub>0</sub> = 0}}, with coefficients {{math|''f''<sub>''n''</sub>, ..., ''f''<sub>0</sub> ∈ ''F'', ''n'' > 0}},
has a solution {{math|''x'' ∊ ''F''}}.<ref>{{harvp|Artin|1991|loc=§13.9}}</ref> By the [[fundamental theorem of algebra]], {{math|'''C'''}} is algebraically closed, i.e., ''any'' polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are ''not'' algebraically closed since the equation
: {{math|1=''x''<sup>2</sup> + 1 = 0}}
does not have any rational or real solution. A field containing {{math|''F''}} is called an ''[[algebraic closure]]'' of {{math|''F''}} if it is [[algebraic extension|algebraic]] over {{math|''F''}} (roughly speaking, not too big compared to {{math|''F''}}) and is algebraically closed (big enough to contain solutions of all polynomial equations).

By the above, {{math|'''C'''}} is an algebraic closure of {{math|'''R'''}}. The situation that the algebraic closure is a finite extension of the field {{math|''F''}} is quite special: by the [[Artin–Schreier theorem]], the degree of this extension is necessarily {{math|2}}, and {{math|''F''}} is [[elementarily equivalent]] to {{math|'''R'''}}. Such fields are also known as [[real closed field]]s.

Any field {{math|''F''}} has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as ''the'' algebraic closure and denoted {{math|{{overline|''F''}}}}. For example, the algebraic closure {{math|{{Overline|'''Q'''}}}} of {{math|'''Q'''}} is called the field of [[algebraic number]]s. The field {{math|{{overline|''F''}}}} is usually rather implicit since its construction requires the [[ultrafilter lemma]], a set-theoretic axiom that is weaker than the [[axiom of choice]].<ref>{{harvp|Banaschewski|1992}}. [https://mathoverflow.net/questions/46566/is-the-statement-that-every-field-has-an-algebraic-closure-known-to-be-equivalent Mathoverflow post]</ref> In this regard, the algebraic closure of {{math|'''F'''<sub>''q''</sub>}}, is exceptionally simple. It is the union of the finite fields containing {{math|'''F'''<sub>''q''</sub>}} (the ones of order {{math|''q''<sup>''n''</sup>}}). For any algebraically closed field {{math|''F''}} of characteristic {{math|0}}, the algebraic closure of the field {{math|''F''((''t''))}} of [[Laurent series]] is the field of [[Puiseux series]], obtained by adjoining roots of {{math|''t''}}.<ref>{{harvp|Ribenboim|1999|loc=p. 186, §7.1}}</ref>