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