s there any 'explicit' proof, preferably without the use of cohomology, of second inequality of global field?
By second inequality I mean $[C_K:N_{L/K}C_L]\leq [L:K]$ where $C_K$ is idele class group of $K$. In particular I would like to know proof for the case where char($K)= [L:K]=p$ where $p$ is a prime.
I know in the book Class field theory by Artin and Tate there is a proof but wondering whehter there are more explicit proof, using artin-schreier extension or something like that.
The proof I know of (not necessarily understand it well)
Analytic proof using L-function and so forth (I have practically no knowledge on them and at the very least I am not too interested in this approach for now)
Artin and Tate- It must be a good method, since every text which I found which does not deal with $p$-extensions of function field (which is practically all the text), but I am really struggling to understand it well. Nothing wrong with this method, but I was hoping there is a 'better' approach somehow as it is a fairly old text.
Class Formation II by Kawada Satake- Uses Duality using Witt vectors and actually proves that for any characteristic $p$ field, there is such duality and certain object defines a class formation. And using this, class field theory restricted to $p$-extension of function field can be proved and second inequality is proved on the way. However from what I can tell it relies on Cohomology heavily and ideally I want to see a proof without it. (Like what Neukirch did)
Cross posted at https://mathoverflow.net/questions/195715/explicit-description-calculation-of-norm-group-of-ideles-of-characteristic-p-g