Editing Kronecker–Weber theorem

{{short description|Every finite abelian extension of Q is contained within some cyclotomic field}}
In [[algebraic number theory]], it can be shown that every [[cyclotomic field]] is an [[abelian extension]] of the [[rational number field]] '''Q''', having [[Galois group]] of the form [[modular arithmetic|<math>(\mathbb Z/n\mathbb Z)^\times</math>]]. The '''Kronecker–Weber theorem''' provides a partial converse: every finite abelian extension of '''Q''' is contained within some cyclotomic field. In other words, every [[algebraic integer]] whose [[Galois group]] is [[abelian group|abelian]] can be expressed as a sum of [[root of unity|roots of unity]] with rational coefficients. For example,
:<math>\sqrt{5} = e^{2 \pi i / 5} - e^{4 \pi i / 5} - e^{6 \pi i / 5} + e^{8 \pi i / 5},</math> <math>\sqrt{-3} = e^{2 \pi i / 3} - e^{4 \pi i / 3},</math> and <math>\sqrt{3} = e^{\pi i / 6} - e^{5 \pi i / 6}.</math>
The theorem is named after [[Leopold Kronecker]] and [[Heinrich Martin Weber]].

==Field-theoretic formulation==
The Kronecker–Weber theorem can be stated in terms of [[field (mathematics)|fields]] and [[field extension]]s.
Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers '''Q''' is a subfield of a cyclotomic field.
That is, whenever an [[algebraic number field]] has a Galois group over '''Q''' that is an [[abelian group]], the field is a subfield of a field obtained by adjoining a [[root of unity]] to the rational numbers.

For a given abelian extension ''K'' of '''Q''' there is  a ''minimal'' cyclotomic field that contains it. The theorem allows one to define the [[conductor (algebraic number theory)|conductor]] of ''K'' as the smallest integer ''n'' such that ''K'' lies inside the field generated by the ''n''-th roots of unity. For example the [[quadratic field]]s have as conductor the [[absolute value]] of their [[Discriminant of an algebraic number field|discriminant]], a fact generalised in [[class field theory]].

==History==
The theorem was first stated by {{harvs|txt|authorlink=Leopold Kronecker|last=Kronecker|year=1853}} though his argument  was not complete for extensions of degree a power of 2. 
{{harvs|txt|authorlink=Heinrich Martin Weber|last=Weber|year=1886}} published a proof, but this had some gaps and errors that were pointed out and corrected by {{harvtxt|Neumann|1981}}. The first complete proof was given by {{harvs|txt|last=Hilbert|authorlink=David Hilbert|year=1896}}.

==Generalizations==
{{harvs|txt|last1=Lubin|last2=Tate|year1=1965|year2=1966}} proved the local Kronecker–Weber theorem which states that any abelian extension of a [[local field]] can be constructed using cyclotomic extensions and [[Lubin–Tate extension]]s.   {{harvs|txt|last=Hazewinkel|year=1975}}, {{harvs|txt|last=Rosen|year=1981}} and {{harvs|txt|last=Lubin|year=1981}} gave other proofs.

[[Hilbert's twelfth problem]] asks for generalizations of the Kronecker–Weber theorem to base fields other than the rational numbers, and asks for the analogues of the roots of unity for those fields. A different approach to abelian extensions is given by [[class field theory]].

==References==
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==External links==
{{wikisource|de:David Hilbert Gesammelte Abhandlungen Erster Band – Zahlentheorie/Kapitel 6|Ein neuer Beweis des Kroneckerschen Fundamentalsatzes über Abelsche Zahlkörper.}}

{{DEFAULTSORT:Kronecker-Weber theorem}}
[[Category:Class field theory]]
[[Category:Cyclotomic fields]]
[[Category:Theorems in algebraic number theory]]