Kronecker–Weber theorem

The Kronecker–Weber theorem states that every finite abelian extension of the rational numbers is contained in a cyclotomic field, a field obtained by adjoining a root of unity.

Cyclotomic field: A field obtained by adjoining a root of unity to the rational numbers. The theorem places every finite abelian extension inside one of these fields.
Field extension: A field containing a smaller field as a subfield. The theorem classifies a particular family of finite extensions of the rationals.
Leopold Kronecker: A German mathematician who made foundational contributions to number theory and algebra. His work helped establish the theorem's vision of generating abelian extensions using roots of unity.
Quadratic field: A degree-two extension of the rational numbers. Every quadratic field is abelian and therefore lies in a suitable cyclotomic field.
Nonabelian extension: A Galois field extension whose Galois group is noncommutative. Such extensions fall outside the theorem's hypothesis and need not be cyclotomic subfields.
Conductor (number theory): The least modulus needed to describe an abelian number field through Dirichlet characters. The conductor identifies a cyclotomic field large enough to contain the extension.
Abelian group: A group whose operation is commutative. The extension qualifies precisely when its Galois group is abelian.
Heinrich Martin Weber: A German mathematician known for work in algebra, number theory, and mathematical education. He gave a proof of the theorem that became central to its accepted form.
Quadratic reciprocity: A theorem describing when one prime is a square modulo another. Cyclotomic fields provide a field-theoretic setting for reciprocity phenomena involving quadratic extensions.
Abelian class field theory: The theory classifying abelian extensions of number fields through arithmetic invariants of the base field. The theorem is its particularly explicit and complete case over ℚ.

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