Local classfield theory
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In mathematics, local classfield theory is the study in number theory of the abelian extensions of local fields. It is in itself a rather successful theory, leading to definite conclusions. It is also important for (and was developed to help elucidate) the proofs of class field theory itself.
The basic theory concerns for a local field K, the description of the Galois group G of the maximal abelian extension of K. This is closely related to K×, the multiplicative group of K\{0}. These groups cannot be equal: as a topological group G is pro-finite and so compact. On the other hand K× is not compact.
Taking the case where K is a finite extension of the p-adic numbers Qp, we can say more precisely that K× has the structure of a cartesian product of a compact group with an infinite cyclic group. The main topological operation is to replace the infinite cyclic group by a group Z^, i.e. its pro-finite completion with respect to subgroups of finite index. This can be done by indicating a topology on K×, for which we can complete. This, roughly speaking, is then the correct group to identify with G.
The actual isomorphism used is important in practice, and is described in the theory of the norm residue symbol.