Kronecker–Weber theorem

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In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root of unity to the rational numbers.

Kronecker provided most of the proof in 1853, with Weber in 1886 and Hilbert in 1896 filling in the gaps. It can be proven by a straightforward algebraic construction, though it is also an easy consequence of class field theory and can be proven by putting together local data over the p-adic fields for each prime p.

For a given abelian extension K of Q there is in fact a minimal cyclotomic field that contains it. The theorem allows one to define the 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 fields have as conductor the absolute value of their discriminant, a fact broadly generalised in class field theory.