I was going through Silverman's Advanced topics in the arithmetic of elliptic curves, and this statement was given without proof. Any hints/clarification would be appreciated. He first introduces the $\textit{Global Artin Homomorphism}$, where $L|K$ is a finite abelian extension of global fields, as $$ \theta_{K}^{L}: \mathbb{I}_{K} \mapsto \text{Gal}(L|K), \quad \quad (a_{\nu}) \mapsto \prod_{\nu}\phi_w\circ\theta_{K_{\nu}}^{L_{w}}(a_{\nu})$$
Here $\mathbb{I}_{K}$ denotes $\mathbb{A}_{K}^{*}$ (the idele group), $\theta_{K_{\nu}}^{L_{w}}$ is the composition as follows: $ K_{\nu}^{*} \mapsto \text{Gal}(K_{\nu}^{ab}|K_{\nu}) \mapsto \text{Gal}(L_{w}|K_{\nu})$, where first map is the local reciprocity map, followed by projection. Finally $\phi_{w}: \text{Gal}(L_{w}|K_{\nu}) \mapsto \text{Gal}(L|K)$ is given by $\sigma \mapsto \sigma|_{L}$.
The product takes place in $\text{Gal}(L|K)$, and it is well-defined, since most terms of the product are the identity (as most entries of $(a_{\nu})$ are units). Maybe it's worth mentioning that $w$ is an absolute value that extends $\nu$, and $K_{\nu}$ the completion of $K$ w.r.t to $\nu$ and similarly for $L_{w}$.
So far, so good. He then claims that for $s=(s_{\mathfrak p})$, an idele whose ideal $(s)= \prod_ {\mathfrak p} \mathfrak p^{ord(s_{\mathfrak p})}$ is such that this fractional ideal is not divisible by any primes that ramify in $L$, then $\theta_{K}^{L}(s_{\mathfrak p}) = \bigg( \frac{L|K}{\mathfrak (s)} \bigg)$, where $\bigg( \frac{L|K}{} \bigg)$ dentoes the Artin map, but this has confused me a bit.
If $(s)= \prod_ {\mathfrak p} \mathfrak p^{\nu_{p}}$, then $\bigg( \frac{L|K}{(s)} \bigg) = \prod_{\mathfrak p} \bigg(\frac{L|K}{\mathfrak p} \bigg)^{\nu_{p}}$ since each $\mathfrak p$ is unramified in $L$. Therefore, to show $\theta_{K}^{L}(s_{\mathfrak p}) = \bigg( \frac{L|K}{\mathfrak (s)} \bigg)$, we only need to show $\phi_{w}\circ \theta_{K_{\nu}}^{L_{w}}(\mathfrak p) $ is $\bigg( \frac{L|K}{\mathfrak p} \bigg)$. But this last equality is not too clear to me.