Inalgebraic number theory, it can be shown that everycyclotomic field is anabelian extension of therational number fieldQ, havingGalois group of the form${\displaystyle (\mathbb{Z}/n\mathbb{Z}{)}^{\times }}$. TheKronecker–Weber theorem provides a partial converse: every finite abelian extension ofQ is contained within some cyclotomic field. In other words, everyalgebraic integer whoseGalois group isabelian can be expressed as a sum ofroots of unity with rational coefficients. For example,

${\displaystyle \sqrt{5}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5},}$${\displaystyle \sqrt{-3}=e^{2\pi i/3}-e^{4\pi i/3},}$ and${\displaystyle \sqrt{3}=e^{\pi i/6}-e^{5\pi i/6}.}$

The theorem is named afterLeopold Kronecker andHeinrich Martin Weber.

The Kronecker–Weber theorem can be stated in terms offields andfield extensions.Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbersQ is a subfield of a cyclotomic field.That is, whenever analgebraic number field has a Galois group overQ that is anabelian group, the field is a subfield of a field obtained by adjoining aroot of unity to the rational numbers.

For a given abelian extensionK ofQ there is aminimal cyclotomic field that contains it. The theorem allows one to define theconductor ofK as the smallest integern such thatK lies inside the field generated by then-th roots of unity. For example thequadratic fields have as conductor theabsolute value of theirdiscriminant, a fact generalised inclass field theory.

The theorem was first stated byKronecker (1853) though his argument was not complete for extensions of degree a power of 2.Weber (1886) published a proof, but this had some gaps and errors that were pointed out and corrected byNeumann (1981). The first complete proof was given byHilbert (1896).

Lubin and Tate (1965,1966) proved the local Kronecker–Weber theorem which states that any abelian extension of alocal field can be constructed using cyclotomic extensions andLubin–Tate extensions. Hazewinkel (1975), Rosen (1981) and Lubin (1981) gave other proofs.

Hilbert's twelfth problem asks for generalizations of the Kronecker–Weber theorem to describe abelian extensions of arbitrary number field and asks for the analogues of the roots of unity for those fields. Currently, only forCM-fields are proven generalizations.

A different approach to abelian extensions is given byclass field theory.

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EnglishWikisource has original text related to this article:

Ein neuer Beweis des Kroneckerschen Fundamentalsatzes über Abelsche Zahlkörper.

• Class field theory
• Cyclotomic fields
• Theorems in algebraic number theory

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