The reciprocity map in the local case can be motivated by considering the unramified case. Let me try to explain this. For simplicity, let us consider the case of a finite abelian unramified extension $K_n/bf Q_p$ of degree $n$. Denote by $k_n$ the residue field of $K_n$. The unramified condition gives rise to the isomorphism
$$Gal(K_n/ {bf Q_p}) simeq Gal(k_n / {bf F_p}) simeq {bf Z}/n.$$
Now we like to parametrize these abelian unramified extensions {$ K_n$} of $bf Q_p$ using information from $bf Q_p$. However, $bf Q_p$ is not finitely generated as a module over $bf Z_p$, let along over $bf Z$. $bf Q_p^times$, on the other hand, decomposes as
$$bf Q_p^times simeq p^{bf Z} times {bf Z}_p^times simeq bf Z times mu_{p-1} times Z_p,$$
which, if anything, at least contains a copy of $bf Z$ (depending on the choice of a uniformizer, say $p$).
It then seems somewhat natural to consider the map $${bf Q_p^times} xrightarrow{Art} Gal({bf Q^{ab, un}_p}/{bf Q_p}) qquad p mapsto Frob$$
where $Frob$ is a choice of a topological generator for $Gal({bf Q_p^{ab, un}}/{bf Q_p})$, the Galois group of the maximal abelian unramified extension ${bf Q_p^{ab, un}}$ of $bf Q_p$, which is naturally isomorphic to $Gal(bar{bf F_p} / bf F_p)$ (which is itself non-canonically isomorphic to $hat {bf Z}$).
Now compose the Artin map $Art$ with the restriction map $Gal({bf Q_p^{ab, un}}/{bf Q_p}) to Gal(K_n/ {bf Q_p})$, we obtain a map
$${bf Q_p}^times xrightarrow{Art_n} Gal(K_n/ {bf Q_p})$$ whose kernel is $$p^{n {bf Z}} times bf Z_p^times,$$ which coincidentally is also the image of the norm of $K_n^times$ in $bf Q_p^times$.
This also sheds some light on the global situation, where you have Frobenius at all but finitely many primes. Don't quote me on this, but I recall the Artin reciprocity map is uniquely determined by its action on the Frobenii (I assume a Chebatorev density argument will show this, and you can prove Chebatorev Density Theorem independent of class field theory if I remember correctly.)
Lastly, we saw that the (local) reciprocity map depends on the choice of a Frobenius as well as that of a uniformizer.
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