As noted earlier, I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly rigorous and clean.
In any case, the second part of the book is "thoroughly unmodern", in the author's own terms. He dispenses with cohomology and builds the whole theory based on central simple algebras. And he advises the reader to make it an exercise to himself for working out the "hidden cohomology" in it.
I personally felt this as some kind of perversion(just as in the fixation for Haar measure in the first part) to show the reader that he is a powerful mathematician could do things precisely the way he wants, and my feeling was strengthened by the title "Basic Number Theory".
However since he is a great mathematician, and I am nobody, I feel unqualified dismiss him like that. There must be some merits/uses of doing it this way, rather than using cohomology. If somebody can enlighten me, I would be very grateful.
Again, is it the case that classfield theory can be done with just $H^1$? I wanted to study the subject and attempted, but was put off for a long time by the wrong book choice, reading Weil which felt like drinking stone soup, and it takes me time to take up the subject again.