I am currently reading the book "Algebraic number theory" published by Cassels and Fröhlich. Let $K$ be a global field (which we may think as a number field as far as I am concerned) and $L/K$ a cyclic extension of degree $n\geq 1$. Normalized valuations of $K$ (resp. of $L$) are called primes. If a prime $w$ of $L$ restricts to a prime $v$ of $K$, we denote it by $w|v$. Given a prime $w$ of $L$, we denote by $L_w$ its completion with respect to $w$, and $U_w$ the group of units of $L_w$.
In the process of proving that the Herbrand quotient of the ideal class group $C_L$ of $L$ is $n$ (Chapter VII, 8.3 Theorem p.178), Tate first starts arguing the following way:
Denote by $J_L$ the group of idèles of $L$. There exists a finite set $S$ of primes of $K$ so large that we have $J_L=L^{\star}J_{L,S}$ where $$J_{L,S}=\prod_{v\in S}\left(\prod_{w|v}L^{\star}_w\right)\times \prod_{v\not\in S}\left(\prod_{w|v}U_w\right)$$
To be more precise, Tate adds that « the set $S$ must include the archimedian primes of $K$, the primes of $K$ ramified in $L$ and all primes of $K$ which lie below some primes whose classes generate the ideal class group of $L$. »
Now, I may be just confused at the moment, but I fail to see the reason why this assertion holds.
First, I understand that there are only a finite number of primes of $K$ satisfying one of the first two conditions, but what about the third ?
Secondly, given an $S$ as prescribed above, why can we decompose any idèle as a product of non-zero element of $L$ and an idèle belonging to $J_{L,S}$ ?
I do know that $J_L$ can be seen as the direct limit of the $J_{L,S}$ for $S$ ranging on those finite sets of primes containing the archimedian and the ramified ones, but I fail to relate this fact to the desired decomposition.
Any help on this matter would be greatly appreciated.