Kronecker–Weber theorem
Every finite abelian extension of Q is contained within some cyclotomic field / From Wikipedia, the free encyclopedia
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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 . 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 can be expressed as a sum of roots of unity with rational coefficients. For example,
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The theorem is named after Leopold Kronecker and Heinrich Martin Weber.