Class Field Theory

These12 are the notes for Math 776, University of Michigan, Winter 1997, slightly revised from those handed out during the course. They have been substantially revised and expanded from an earlier version, based on my notes from 1993 (v2.01).My approach to class field theory in these notes is eclectic. Although it is possible to 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 cohomology groups associated with number fields. When it sheds additional light, I have not hesitated to include more than one proof of a result.The heart of the course is the odd numbered chapters. Chapter II, which is on the cohomology of groups, is basic for the rest of the course, but Chapters IV, VI, and VIII are not essential for reading Chapters III, V, and VII. Except for its first section, Chapter I can be skipped by those not interested in explicit local class field theory.