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I am interested to learn class field theory. I want to know whether or not group homology and cohomology are prerequisites for class field theory. One of the answers to this question suggests that one can learn local class field theory without homology and cohomology.

Is it possible to learn global class field theory without homology and cohomology ?

Also, what would be a good book (or lecture note) to study homology and cohomoogy (suitable for beginners) ?

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    $\begingroup$ For homology and cohomology I would recommend Algebraic topology by Hatcher. $\endgroup$
    – EHH
    Aug 30, 2016 at 11:06
  • $\begingroup$ @EHH But there is no group (co)homology in that book. $\endgroup$
    – Danu
    Aug 30, 2016 at 11:30
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    $\begingroup$ J.S. Milne's notes on class field theory use group cohomology to do local class field theory (before doing global cft), and give all the required background on group cohomology. $\endgroup$
    – Terry
    Aug 30, 2016 at 11:32
  • $\begingroup$ @Danu yes you are right about this. I was answering just the final question about books for homology or cohomology which are suitable for beginners. Also when studying this subject I found it easier to start from geometric and topologically inspired angles before moving to abstract homological algebra. But your right that the OP might want to start from somewhere more aligned with their area of interest. $\endgroup$
    – EHH
    Aug 30, 2016 at 11:58

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Elementary books on class field theory are Janusz and Childress; see also Garbanati's beautiful article (http://projecteuclid.org/euclid.rmjm/1250128658). For cohomology in a number theoretical context I strongly suggest Weiss (Cohomology of groups).

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  • $\begingroup$ Are you talking about the book Algebraic Number Fields by Janusz ? $\endgroup$
    – learning_math
    Sep 27, 2016 at 14:51
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    $\begingroup$ Yes. For those who can read German, Hasse's lectures are worth looking at, and there's a readable introduction in French by Bijakowski. $\endgroup$
    – franz lemmermeyer
    Sep 27, 2016 at 16:33

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