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Kronecker–Weber theorem
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{{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]].
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