= Solution
The main theorem of <local class field theory> consists of <Local Artin reciprocity> and its existence theorem. There is a canonical continuous reciprocity map
$$
\operatorname{Art}_K:K^\times\longrightarrow\operatorname{Gal}(K^{\mathrm{ab}}/K)
$$
with dense image, normalized so that a <uniformizer> maps to a chosen <Frobenius>. For every finite abelian extension $L/K$, it induces an isomorphism
$$
\boxed{K^\times/N_{L/K}(L^\times)\cong\operatorname{Gal}(L/K).}
$$
Moreover, 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 $K^\times$.
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