Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 123 / 4 / ii / Solution

The main theorem of local class field theory consists of Local Artin reciprocity and its existence theorem. There is a canonical continuous reciprocity mapwith dense image, normalized so that a uniformizer maps to a chosen Frobenius. For every finite abelian extension $L/K$, it induces an isomorphismMoreover, the existence theorem of local class field theory says that $L\mapsto N_{L/K}(L^{\times })$ is an inclusion-reversing bijection between finite abelian extensions of $K$ in $K^{\mathrm{ab}}$ and finite-index open subgroups of $K^{\times }$.