maximal abelian extension

Among all the abelian extensions of a field — the ones with commuting symmetries — there is a single largest one that contains them all at once. The maximal abelian extension is that universal container: pile up every abelian extension you can, and their union is again abelian and can grow no further while staying abelian.

Formally, the maximal abelian extension K^ab of a field K is the compositum (inside a fixed algebraic closure) of all finite abelian extensions of K. It is a Galois extension of K, and its Galois group Gal(K^ab/K) is the maximal abelian quotient of the absolute Galois group Gal(K-bar/K), namely the latter's abelianization (the quotient by the closure of its commutator subgroup). Every abelian extension of K is contained in K^ab.

Describing K^ab intrinsically is the central goal of class field theory. For K = Q the answer is the Kronecker-Weber theorem: Q^ab is the union of all cyclotomic fields Q(zeta_n), so Gal(Q^ab/Q) is the unit group of the profinite integers. For general number fields the description is via the idele class group and the Artin reciprocity map; finding an explicit analogue of roots of unity for arbitrary number fields is the content of Hilbert's twelfth problem, still only partially resolved.