CLASS FIELD THEORY - James Milne

CLASS FIELD the notes for Math 776, University of Michigan, Winter 1997, slightlyrevised from those handed out during the course. They have been substantiallyrevised and expanded from an earlier version, based on my notes from 1993 ( ).My approach to CLASS FIELD THEORY in these notes is eclectic. Although it is possibleto prove the main theorems in CLASS FIELD THEORY using neither analysis nor cohomology,there are major theorems that can not even be stated without using one or the other,for example, theorems on densities of primes, or theorems about the cohomologygroups associated with number fields. When it sheds additional light, I have nothesitated to include more than one proof of a heart of the course is the odd numbered chapters. Chapter II, which is on thecohomology of groups, is basic for the rest of the course, but Chapters IV, VI, andVIII are not essential for reading Chapters III, V, and VII.

iii. Weil,A.,BasicNumberTheory,Springer,1967. ThearticlesofSerreandTateinCasselsandFr¨ohlich1967,andJanusz1973,have beenespeciallyusefulinthewritingofthesenotes.

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