Reciprocity map in global class field theory 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 $v$:
$$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\rm ab}/K_v)$$
sending $\varpi_v$, a generator of the copy of $\mathbf{Z}$ in $K_v^{\times}$, to the Frobenius element in $\text{Gal}(K_v^{\rm unr}/K_v)$, and sending $\mathcal{O}_v^{\times}$ isomorphically onto $\text{Gal}(K_v^{\rm LT}/K_v)$, where $K_v^{\rm LT}$ is the Lubin-Tate extension of $K_v$, with $K_v^{\rm LT}\cdot K_v^{\rm unr} = K_v^{\rm ab}$.
The completion map $K\to K_v$ induces an injective map $$\alpha_v : \text{Gal}(K_v^{\rm ab}/K_v)\to\text{Gal}(K^{\rm ab}/K)$$
Let's call $c_v$ the map $K^{\times}\to K_v^{\times}$ induced by completion.
Is there an explicit description for:
$$R_v := \alpha_v\circ\rho_v\circ c_v : K^{\times}\to\text{Gal}(K^{\rm ab}/K)\ ?$$
 A: In view of the so called "global-local principle", it does not seem very natural to consider only one single place $v$ of $K$. Usually, one fixes a rational prime $p$ and the set $S$ of all places of $K$ above $p$, and one introduces the maximal abelian extension $K_S$ of $K$ which is unramified outside $S$, as well as the subfield $H_S$ of $K_S$ in which all the places of $S$ are completely decomposed.  Note that $H_S$ is a modified Hilbert class field, with $Gal(H_S /K) \cong Cl_S (K)$, the quotient of the class group of $K$ modulo the subgroup generated by the classes of the primes in $S$. Then CFT (in idelic terms) gives a natural exact sequence $1 \to \bar E_S \to \prod_S Gal({K_v}^{ab}/K_v) \to Gal(K_S /K) \to Gal(H_S /K) \to 1$. The leftmost term is the closure of the group of $S$-units of $K$ embedded diagonally in the product of the profinite completions $\bar{K_v}^*$ of the groups ${K_v}^*, v\in S$, each $\bar{K_v}^*$ being $\cong Gal({K_v}^{ab}/K_v)$ via $\rho_v$. If we wave hands and say that we "know" the two extreme terms, then we "know"  what you call the maps $R_v$ provided we know the local $\rho_v$'s ("local reciprocity laws" of Hasse, Brückner, Kato, etc.). Note that your maps $\alpha_v$ cannot be injective in this setting.   
