KroneckerWeber
The Kronecker–Weber theorem states that every finite abelian extension of the rational numbers is contained in a cyclotomic field Q(ζ_n) for some n, where ζ_n is a primitive nth root of unity. Equivalently, every finite abelian extension of Q is a subfield of some cyclotomic field. A cyclotomic field is obtained by adjoining a root of unity to the rational numbers.
Consequences and description of Galois groups. In particular, the maximal abelian extension of Q is the union
Historical context and significance. The conjecture was formulated by Leopold Kronecker and proved by Heinrich Weber