abelian extension

An abelian extension is a Galois extension whose symmetries all commute with each other: applying two of them in either order gives the same result. Commuting symmetries are vastly easier to organize than tangled non-commuting ones, which is why abelian extensions are the part of Galois theory we understand most completely.

Formally, a Galois extension L/K is abelian if the Galois group Gal(L/K) is an abelian group. Because subgroups of abelian groups are automatically normal, the Galois correspondence becomes especially clean: every intermediate field is itself Galois over K, and the lattice of subfields mirrors the (well-understood) subgroup lattice of an abelian group. Cyclotomic extensions and Kummer extensions are the prototypical examples.

Abelian extensions are the subject of class field theory, one of the deep achievements of twentieth-century number theory, which classifies all abelian extensions of a given number field in terms of intrinsic arithmetic data (ideals, ideles, ray class groups). The Kronecker-Weber theorem is the prototype: every abelian extension of Q lies inside a cyclotomic field, so the maximal abelian extension of Q is the union of all Q(zeta_n).