I posted the same question earlier in stack exchange, (https://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field) thinking it is most definitely not a research level question but I have not got a satisfactory answer so I'll try here.
Ultimately I am interested in the calculation of $[C_K:N_{L/K}C_L]\leq [L:K]$ where $C_K$ is idele class group of $K$ where char($K)= [L:K]=p$ where $p$ is a prime and $K$ is a global field. In the number field case we have explicit description of norm group, which can be found in many text, including Artin-Tate, Neukirch, Cassels-Frohlich. In the case of function field case, however, I have only found one text which is relevant, namely Artin-Tate. As I have not completely understood this proof yet so I will not comment on it too much but it does not describe $C_K/N_{L/K}C_L$ in an explicit manner.
So to restate my question, is there a way to write down $C_K/N_{L/K}C_L$ in the setting I am talking about? Or is there any alternative (algebraic) method of proving $[C_K:N_{L/K}C_L]\leq p$?
Update:
I have now read, and understood the proof in Artin-Tate and I think it's neat and serves my purpose. However, it would still be nice to have some explicit description of $C_K/N_{L/K}C_L$ in the setting I am talking about. I would be genuinely quite surprised to see that it has not been done before but I am yet to find any source which describes it. Or unlike my expectation has this not been done?
