A Brief Summary of the Statements of Class Field Theory

where U ranges over the finite-index open normal subgroups of G. There is a natural continuous homomorphism G → Ĝ through which every other continuous homomorphism from G to a profinite group factors uniquely. If G is profinite already, then G → Ĝ is an isomorphism. In general, G→ Ĝ need not be injective or surjective. Nevertheless, we think of G as being almost isomorphic to Ĝ: The finite-index open subgroups of G are in bijection with those of Ĝ. And finite-index open subgroups of certain Galois groups are what we are interested in. . .