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Background (the Kummer extension case) Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested in $R=...

Part I. Say $K$ is a number field, $v$ is a finite place of $K$, $K_v$ the $v$-adic completion of $K$. We have the local Artin map for every finite $v$: $$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\...