# Group cohomology or classical approach for class field theory?

First of all, I don't think this is a duplicate, because the related questions I found were mainly about history of group cohomology in number theory and there was no one asking about the classical approach versus the cohomological approach. If you find one, post it below and I'll consider deleting this one. Thanks.

I'm an $2$nd year undergraduate student currently studying algebraic number theory by Neukirch's book and I think that I covered most of the things people usually consider the basics, which is the study of algebraic number fields, integrality, the ideal class group, ramification of prime ideals and the theory of $p$-adic numbers and valuations in general. It feels like the next step is (and I think that I have more or less appropriate background) to dive in class field theory.

Doing a little search, I found that class field theory (both local and global) can be done in a variety of ways, for example using group cohomology. I don't know anything about group cohomology, but I think that Neukirch's approach does not use group cohomology, even though he talks about groups $H^{0}(G(L/K),A_{L})$, which I believe is notation that shows up in group cohomology.

1. In the aspect of difficulty, which approach should I take? Neukirch vs. Group cohomology (for example in Milne's online notes for CFT)

2. If group cohomology was the answer for the previous one, then should I learn some category theory before group cohomology or it is not that related?

3. Are the two approaches actually distinct or just different in language? For example, is the the approach via group cohomology just identifying number-theoretical objects with "cohomological" objects and then applying theorems?

4. Is it worthy to try and do both at the same time and get a wide view on the subject?

EDIT: I forgot that Neukirch has a book about class field theory only. I'm actually using his Algebraic Number Theory.

• I don’t think you need to know much category theory to do group cohomology. (Others may disagree.) – Lubin Dec 23 '15 at 5:06