Local class field theory

Summary

Local class field theory describes the abelian extensions of a non-Archimedean local field $K$ in terms of the multiplicative group $K^\times$. Its main result is the local reciprocity map $K^\times \to \Gal(K^{ab} / K)$, where $K^{ab}$ is the maximal abelian extension of $K$. For each finite abelian extension $L / K$, it induces an isomorphism $K^\times / N_{L/K}(L^\times) \cong \Gal(L / K)$. The finite abelian extensions of $K$ correspond to the open subgroups of finite index of $K^\times$, through their norm groups.