1  a brief review  1  number fields  2  completions of number fields  3  some general questions motivating class field theory2 dirichlet's theorem on primes in arithmetic progressions  1  characters of finite abelian groups  2  dirichlet characters  3  dirichlet series  4  dirichlet's theorem on primes in arithmetic progressions  5  dirichlet density3 ray class groups  1  the approximation theorem and infinite primes  2  ray class groups and the universal norm index inequality  3  the main theorems of class field theory4 the idelie theory  1  places of a number field  2  a little topology  3  the group of ide1es of a number field  4  cohomology of finite cyclic groups and the herbrand quotient  5  cyclic galois action on ideles5  artin reciprocity  1  the conductor of an abelian extension of number fields and the artin symbol  2  artin reciprocity  3  an example: quadratic reciprocity  4  some preliminary results about the artin map on local fields6 the existence theorem, consequences and applications  l  the ordering theorem and the reduction lemma  2  kummer n-extensions and the proof of the existence theorem  3  the artin map on local fields  4  the hilbert class field  5  arbitrary finite extensions of number fields  6  infinite extensions and an alternate proof of the existence theorem.  7  an example: cyclotomic fields7  local class field theory  1  some preliminary facts about local fields  2  a fundamental exact sequence  3  local units modulo norms  4  one-dimensional formal group laws  5  the formal group laws of lubin and tare  6  lubin-tate extensions  7  the local artin mapbibliographyindex