= Lubin–Tate description of local reciprocity
{c}
{title2=$\operatorname{Art}_K(\pi^m u)|_{K_\pi}=[u^{-1}],\quad \operatorname{Art}_K(\pi^m u)|_{K^{\mathrm{nr}}}=\operatorname{Frob}_K^m$}
With the <arithmetic Frobenius> normalization, a chosen <uniformizer> $\pi$ and <Lubin–Tate formal group> give these actions on the <Lubin–Tate tower> and the <maximal unramified extension>. The two fields are linearly disjoint and their compositum is the <maximal abelian extension>. The inverse on units is essential: using geometric Frobenius would invert the entire <Local Artin map>. The finite kernels on a compositum of residue degree $d$ and torsion level $r$ are $\pi^{d\mathbb Z}(1+\pi^r\mathcal O_K)$. They are a basis of finite-index open subgroups, yielding the <profinite completion> of $K^\times$ as the abelian <Galois group>.
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