The ideles of $\mathbb{Q}$, lets denote them by $\mathbb{I}$, satisfy the following decomposition:
$\mathbb{Q}^\times\times \mathbb{R}_{>0}\times\prod_p \mathbb{Z}_p^\times \to \mathbb{I}$
The map is given by $(r,t,(u_p))\mapsto (rt,ru_2,ru_3,ru_5,\ldots)$
and it's an isomorphism of topological groups if $\mathbb{Q}^\times$ is given the discrete topology. Now let $K/\mathbb{Q}$ be finite abelian, so that we have the composition of maps given by the reciprocity theorem:
$\mathbb{Q}^\times\times \mathbb{R}_{>0}\times\prod_p \mathbb{Z}_p^\times \to \mathbb{I}\to \textrm{Gal}(K/\mathbb{Q})$
My questions are the following:
How can we identify the kernel of this map on the left side? My guess based on some trivial examples is that if $p$ is unramified, then $\mathbb{Z}_p^\times$ is contained in the kernel. Is this true? If so could anyone show how its done? If $p$ ramfies, can we say anything?
If $K/\mathbb{Q}$ is cyclotomic, then is there a nicer description of the kernel? This would be nice to know, since it would help with identifying a cyclotomic extension that contains a general abelian extension.