I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.

I definitely do not want to avoid homological algebra and Brauer groups in my lectures, but I would prefer to minimize other prerequisites. At the moment I am aware of Cassels and Frolich's book, of Milne's lectures, and of Beilinson's lectures on local class field theory. Are there any books or lecture notes in the internet that differ from these ones significantly? What are the main distinctions?

P.S. Unfortunately, I don't have much time. Yet it would be certainly nice to give some easy applications of the theory. What it the best place to read about them?

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    $\begingroup$ Far from duplicate, but relevant: mathoverflow.net/q/6932/30186 $\endgroup$ – Wojowu Aug 31 at 9:59
  • $\begingroup$ Serge Lang's book on algebraic number theory, and Andre Weil's book "Basic Number Theory" do not really involve Brauer groups explicitly (but implicitly, yes) $\endgroup$ – Venkataramana Aug 31 at 10:06
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    $\begingroup$ Neukirch's book "Class Field Theory" offers a different approach to Cassels-Froehlich. $\endgroup$ – Kevin Buzzard Aug 31 at 21:38
  • $\begingroup$ Janusz’s “Algebraic Number Fields” covers global class field theory using elementary methods and avoids the more cohomological aspects. I admittedly did not like it as much, but you may find it useful. $\endgroup$ – robinz16 Sep 1 at 7:07
  • $\begingroup$ In Lang's Algebraic Number Theory, he gives a nice four-page summary of several different approaches to class field theory. $\endgroup$ – Timothy Chow Sep 1 at 15:50

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