Lubin–Tate description of local reciprocity (${\mathrm{Art}}_K({\pi }^{m}u){|}_{K_{\pi }}=[u^{-1}],{\mathrm{Art}}_K({\pi }^{m}u){|}_{K^{\mathrm{nr}}}={\mathrm{Frob}}_K^m$)

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.