Which properties determine the uniqueness of the local Artin map?

Question

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 $N_{w/v}(K_w^{\ast})$. 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)?

Answer 1

This is the local reciprocity law. It says for any non-archimedian local field $K$, there exists a unique group homomorphism $\phi_K : K^\times \rightarrow \mathrm{Gal}(K^{ab}/K)$ with the property that:

i) If $\pi$ is a uniformizer for $K$, and $L/K$ is a finite unramified extension, then $\phi_K(\pi)|_L$ acts as the Frobenius on $L$.

ii) If $L/K$ is a finite abelian extension, then $\mathrm{Nm}_{L/K}(L^\times)$ is in the kernel of $\phi_K|_L$, and there's an induced isomorphism

$$ \overline{\phi}_K|_L : K^\times/\mathrm{Nm}_{L/K}(L^\times)\rightarrow \mathrm{Gal}(L/K).$$

See for example Milne's notes on class field theory.

One remark is that historically the global theorem was proved first analytically, and the local theorem was deduced from it. The idèlic notation you are using was introduced by Chevalley to give a local algebraic proof and to reverse the order of implications.