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 . It extends to a map from the free abelian group on unramified primes to the group (the fractional ideals of ). 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 induces 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 (which has Galois group ) and the subgroups of . Giving an abelian extension of is exactly the same as giving an integer and a subgroup of . Even more importantly, the reciprocity map is simply . This is a nice generalization of quadratic reciprocity: for , the for a prime depends in a simple way on nothing but .
William Stein 2004-05-06