Kronecker–Weber theorem

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

The theorem is named after Leopold Kronecker and Heinrich Martin Weber.

Field-theoretic formulation

The Kronecker–Weber theorem can be stated in terms of fields and field extensions. 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 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 generalised in class field theory.