KroneckerWeberSatz

The Kronecker–Weber theorem (Kronecker–Weber Satz) is a foundational result in algebraic number theory. It states that every finite abelian extension of the rational numbers Q is contained in a cyclotomic field Q(ζ_n), for some n, where ζ_n is a primitive n-th root of unity.

Equivalently, if K/Q is a finite Galois extension with abelian Galois group, then there exists an n and a subgroup H of (Z/nZ)× such that K is the fixed field of H inside Q(ζ_n). In particular, the Galois group Gal(K/Q) is a quotient of (Z/nZ)×. The integer n is called the conductor of K, and K is the subfield of Q(ζ_n) determined by congruence conditions modulo n.

Consequences of the theorem include that every abelian extension of Q is generated by roots of unity, and the maximal abelian extension of Q is the union of all cyclotomic fields. This provides an explicit description of abelian extensions of Q in contrast to the general situation described by class field theory.

Historical context and significance: the result was established in the 19th century by Leopold Kronecker and Heinrich Weber and is considered a precursor to the broader framework of class field theory. It shows that the abelian portion of the absolute Galois group of Q is governed entirely by cyclotomic fields, yielding a complete and concrete description of such extensions.