Editing Field (mathematics) (section)

=== 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>