Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we can define the local Artin map as the composition $$K_v^{\ast} \rightarrow \mathbb{I}_K \rightarrow \text{Gal}(L/K)$$ as well as show this composition maps onto the decomposition group $\text{Gal}(L/K)_v = \text{Gal}(L_w/K_v)$ with kernel $N_{w/v}(L_w^{\ast})$. But the definition of the local Artin map depends on (somewhat arbitrary) global parameters. What is it about the local Artin map that ensures its uniqueness (so that it doesn't depend on the choice of $L/K$ inducing it)?
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$\begingroup$ See 2.8 of Serre's article in Cassels-Frohlich. $\endgroup$– anonMay 28, 2015 at 12:07
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$\begingroup$ Of course, nowadays (since about 1940), one can develop local class field theory purely locally, and so the question doesn't arise. $\endgroup$– anonMay 28, 2015 at 12:23
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