Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only developed global class field theory, and haven't done much locally. For example, suppose the only way we know of producing the local Artin map is something like:
1 . Find an abelian extension of global fields $L/K$, and places $w/v$, such that $K_v = k, L_w = l$ (this is nontrivial, we need e.g. the existence theorem from global class field theory).
2 . For the global Artin map $J_K \rightarrow Gal(L/K)$, define the local Artin map as the restriction to $K_v^{\ast} = k^{\ast}$
3 . Prove the relevant properties of this local map, for example that it maps onto $Gal(l/k)$, that its kernel is $N_{l/k}(l^{\ast})$, that it maps the prime element of $k$ to the Frobenius when $l/k$ is unramified.
This is done in, e.g. Serge Lang's book. The obvious problem here is that we don't have much in the way of uniqueness for this approach. It's not clear why the local Artin map doesn't depend on the choice of abelian extension $L/K$.
I've asked a question on here before about how we can ensure uniqueness of the local Artin map (Which properties determine the uniqueness of the local Artin map?) when it is defined from the global map, and the answer I received suggested that I look at Serre's portion of Algebraic Number Theory. However, the uniqueness theorem proved there used the fact that there was a well defined local Artin map $k^{\ast} \rightarrow Gal(k^{ab}/k)$ for the maximal abelian extension $k^{ab}$ (already kind of a problem when we're defining the local Artin map globally, one abelian extension of $k$ at a time, rather arbitrarily) and that $k^{ab}$ is equal to a compositum $k^{ur}k^{\pi}$, where $k^{ur}$ is the maximal unramified extension, and $k^{\pi}$ is the fixed field of the subgroup generated by the Artin map applied to a uniformizer $\pi$ of $k$.
Worse, this last assertion about the compositum doesn't seem to be trivial (actually, I've been unable to see where exactly it's proved), and might rely on some nontrivial local class field theory which is developed beforehand. What I'm really looking for is a way to get the "essentials" of local class field theory (the existence of the local Artin map, the bijection between closed subgroups of $k^{\ast}$ and abelian extensions of $k$, the uniqueness of the local Artin map) very quickly and without too much effort after having done the hard work of developing global class field theory. Most of this is done, e.g. in Lang, so the only thing left I want to get is the uniqueness of the local Artin map.
So, I've got the main theorems of global C.F.T. to work with, the equalities $[k^{\ast} : N_{l/k}(l^{\ast})] = [l : k]$ and $[\mathcal O_k^{\ast} : N_{l/k}(\mathcal O_w^{\ast})] = e(l/k)$, and not much else. Is this possible to do what I want very easily? Or is there a lot of machinery with local class field theory that needs to be developed to say anything about uniqueness?
