An important example, the
idele group (ideal element group)
I(
K), is the case of G = GL_1. Here the set of
ideles consists of the invertible adeles; but the topology on the idele group is
not their topology as a subset of the adeles. Instead, considering that GL_1 lies in two-dimensional
affine space as the '
hyperbola' defined parametrically by : \{(t,t^{-1})\}, the topology correctly assigned to the idele group is that induced by inclusion in
A2; composing with a projection, it follows that the ideles carry a
finer topology than the subspace topology from
A. Inside
AN, the product
KN lies as a
discrete subgroup. This means that
G(
K) is a discrete subgroup of
G(
A), also. In the case of the idele group, the
quotient group : I(K)/K^\times \, is the
idele class group. It is closely related to (though larger than) the
ideal class group. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a
compact group; the proof of this is essentially equivalent to the finiteness of the class number. The study of the
Galois cohomology of idele class groups is a central matter in
class field theory.
Characters of the idele class group, now usually called
Hecke characters or Größencharacters, give rise to the most basic class of
L-functions. ==Tamagawa numbers==