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