KroneckerWeber

Kronecker–Weber theorem

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 of all cyclotomic fields, and its Galois group is isomorphic to the profinite completion of the unit group of integers, Gal(Q^ab/Q) ≅ ĤZ^×. For each n, the Galois group Gal(Q(ζ_n)/Q) is isomorphic to (Z/nZ)×, and more generally any finite abelian extension K of Q sits inside Q(ζ_n) for some n, with Gal(K/Q) realized as a quotient of (Z/nZ)×. The conductor of K divides n, tying the field extension to arithmetic invariants.

Historical context and significance. The conjecture was formulated by Leopold Kronecker and proved by Heinrich Weber in the late 19th century. The theorem is a foundational result in algebraic number theory and a central early achievement leading to the development of class field theory. It shows that abelian extensions of Q have a complete description in terms of cyclotomic fields, a paradigm that underpins broader questions about abelian extensions of general number fields and their connections to automorphisms and reciprocity laws. See also cyclotomic fields and class field theory.