Thus the conjugacy class of in is a well-defined function of . For example, if is abelian, then does not depend on the choice of lying over and we obtain a well defined symbol called the Artin symbol. It extends to a homomorphism from the free abelian group on unramified primes to . Class field theory (for ) sets up a natural bijection between abelian Galois extensions of and certain maps from certain subgroups of the group of fractional ideals for . We have just described one direction of this bijection, which associates to an abelian extension the Artin symbol (which is a homomorphism). The Kronecker-Weber theorem asserts that the abelian extensions of are exactly the subfields of the fields , as varies over all positive integers. By Galois theory there is a correspondence between the subfields of the field , which has Galois group , and the subgroups of , so giving an abelian extension of is exactly the same as giving an integer and a subgroup of . The Artin reciprocity map is then .
William Stein 2012-09-24