Editing Field (mathematics) (section)

=== Number theory: global fields ===
[[Global field]]s are in the limelight in [[algebraic number theory]] and [[arithmetic geometry]].
They are, by definition, [[number field]]s (finite extensions of {{math|'''Q'''}}) or function fields over {{math|'''F'''<sub>''q''</sub>}} (finite extensions of {{math|'''F'''<sub>''q''</sub>(''t'')}}). As for local fields, these two types of fields share several similar features, even though they are of characteristic {{math|0}} and positive characteristic, respectively. This [[function field analogy]] can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the [[Riemann hypothesis]] concerning the zeros of the [[Riemann zeta function]] (open as of 2017) can be regarded as being parallel to the [[Weil conjectures]] (proven in 1974 by [[Pierre Deligne]]).

[[File:One5Root.svg|thumb|The fifth roots of unity form a [[regular pentagon]].]]
[[Cyclotomic field]]s are among the most intensely studied number fields. They are of the form {{math|'''Q'''(''ζ''<sub>''n''</sub>)}}, where {{math|''ζ''<sub>''n''</sub>}} is a primitive {{math|''n''}}th [[root of unity]], i.e., a complex number {{math|''ζ''}} that satisfies {{math|1={{itco|''ζ''}}<sup>''n''</sup> = 1}} and {{math|{{itco|''ζ''}}<sup>''m''</sup> ≠ 1}} for all {{math|0 < ''m'' < ''n''}}.<ref>{{harvp|Washington|1997}}</ref> For {{math|''n''}} being a [[regular prime]], [[Ernst Kummer|Kummer]] used cyclotomic fields to prove [[Fermat's Last Theorem]], which asserts the non-existence of rational nonzero solutions to the equation
: {{math|1=''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''n''</sup>}}.

Local fields are completions of global fields. [[Ostrowski's theorem]] asserts that the only completions of {{math|'''Q'''}}, a global field, are the local fields {{math|'''Q'''<sub>''p''</sub>}} and {{math|'''R'''}}. Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called the [[local–global principle]]. For example, the [[Hasse–Minkowski theorem]] reduces the problem of finding rational solutions of quadratic equations to solving these equations in {{math|'''R'''}} and {{math|'''Q'''<sub>''p''</sub>}}, whose solutions can easily be described.<ref>{{harvp|Serre|1996|loc=Chapter IV}}</ref>

Unlike for local fields, the Galois groups of global fields are not known. [[Inverse Galois theory]] studies the (unsolved) problem whether any finite group is the Galois group {{math|Gal(''F''/'''Q''')}} for some number field {{math|''F''}}.<ref>{{harvp|Serre|1992}}</ref> [[Class field theory]] describes the [[abelian extension]]s, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the [[Kronecker–Weber theorem]], describes the maximal abelian {{math|'''Q'''<sup>ab</sup>}} extension of {{math|'''Q'''}}: it is the field
: {{math|'''Q'''(''ζ''<sub>''n''</sub>, ''n'' ≥ 2)}}
obtained by adjoining all primitive {{math|''n''}}th roots of unity. [[Kronecker Jugendtraum|Kronecker's Jugendtraum]] asks for a similarly explicit description of {{math|''F''<sup>ab</sup>}} of general number fields {{math|''F''}}. For [[imaginary quadratic field]]s, <math>F=\mathbf Q(\sqrt{-d})</math>, {{math|''d'' > 0}}, the theory of [[complex multiplication]] describes {{math|''F''<sup>ab</sup>}} using [[elliptic curves]]. For general number fields, no such explicit description is known.