In Childress'Class Field Theory book, section 5.3 is an example of how to deduce the Classical Quadratic Reciprocity Law from the idelic version of Artin's Reciprocity Law.
It begins by constructing an idele $c_{\infty} = (-1, 1....)$ where the $-1$ is in the real coordinate. She desires to decide when is $c_{\infty}$ in the Kernel of the Artin map, which by Artin reciprocity happens if and only if \begin{equation*} c_{\infty} \in \mathbb{Q}^{\times}N_{K/\mathbb{Q}} \end{equation*} for a certain quadratic field $K$.
The point is that she says: this is easily seen to be true if and only if $-1$ is a norm from $K_{\infty}$ to $\mathbb{Q}_{\infty} = \mathbb{R}$.
Later, she says a similar claim when checking finite places, constructing $b_l = (1, 1,..., q,...)$ an idele with a prime $q$ in the $l$-th entry. And again, it becomes knowing when is $b_l$ in the kernel, which is $b_l\in \mathbb{Q}^{\times}N_{K/\mathbb{Q}}$, which she says means $q$ is a norml from $K_p$ to $\mathbb{Q}_p$.
I really don't see what is the logic of the argument. In particular, what I don't see is how do you get rid of the $\mathbb{Q}^{\times}$ in front. Artin tells you the kernel is $\mathbb{Q}^{\times}N_{K/\mathbb{Q}}$ not $N_{K/\mathbb{Q}}$.
How is she removing the factor $\mathbb{Q}$ without affecting the entry she is interested in? I tried to do something with approximation theorem to explicitly evaluate the map, but that seems to tell you $\alpha(c_{\infty})_{\infty}$ is a norm, or $\alpha (b_l)_p$ is a norm, which is not what you want.
What am I missing? Can someone help me D: