I'm reading the second edition of Lang, Algebraic Number Theory, page 221. I quote:

Let $F$ be a local field, i.e. the completion of a number field at an absolute value. Let $L$ be an abelian extension with Galois group $G$. Then there exists a number field $k$ and an abelian extension $K$, with absolute value $v$, such that $$ F = k_v, L = K_v$$ For instance, let $E$ be a number field dense in $L$. Let $K$ be the composite of $\sigma E$ for all $\sigma \in G$. Then $K$ is stable under $G$ and we let $k$ be the fixed field of $G$. It is immediate that $k_v = F$, and of course $K_v = L$.

Ookay, so far so good. Then he drops this gem:

Note that the local Artin map $k_v^{\ast} \rightarrow G(K k_v/k_v)$ is induced by the global map. The consistency property of the global symbol implies that the local map is independent of the global extension $K$ over $k$ chosen such that $K_v = L$ and $k_v = F$.

'Consistency' means that for a bigger abelian extension $M$ of $k$ containing $K$, that the restriction of $(x, M/k)$ to $K$ is $(x, K/k)$. But this doesn't explain at all why the local Artin map is independent of the global parameters. You would need to show that for a different abelian extension $K'/k'$ such that $K'_w = L$ and $k'_w = F$, then $(x, K'/k')$ and $(y, K/k)$ can be identified as the same element of $G$ for $x, y$ suitably identified in $k'$ and $k$. Any help here?

P.S. I actually really like Serge Lang's treatment of ANT, loved his complex analysis textbook, it's just frustrating at parts because he assumes you're a Level 99 Clever Warlord.