Today we’ll start reformulating class field theory in the idelic setting. Let 
be a global field (a number field, or a finite separable extension of 
). Let 
be all the places and 
the normalized absolute value at 
. Define 
be to be the completion of at 
. Let 
the ring of integers if is non-archimedean and just itself otherwise.
We’re going to assume that a first algebraic number theory course covers some of the basics of the ideles. Define 
to be the ideles of . Recall this is by definition 
. This is a multiplicative group, but moreover there is a natural topology on it from the restricted direct product. We’ll recall properties of this as needed. We should quickly remind the reader that this is not the same as the product topology. It is the topology generated by the local basis around 
of the form where you take finitely many components to be open sets and the rest must be 
.
The ring of adeles is exactly analogous. We take 
. This is a locally compact topological ring. By construction 
as a set. But the subspace topology is weaker than the topology we give to . You do recover the right topology by taking the subspace of the product 
by 
.
We define now the idele class group to be 
. Fix a conductor 
. Our first theorem is: there is a surjection 
where 
is constructed as follows. Given 
use the approximation theorem to find a non-zero 
with the property that 
if 
and 
where 
ranges through 
for 
. Define 
to be the fractional ideal 
such that 
for all non-archimedean .
Note that if we do this, at very least we’ve constructed a fractional ideal that is relatively prime to 
. Thus we have a candidate map. We should check well-definedness, but that is more tedious than it’s worth. Let’s just take the theorem as true. Out of it we get the nice corollary that there is a surjection 
where 
is the maximal abelian extension of unramified outside . The equality comes from piecing together the Artin maps using class field theory. The kernel is the connected component of the identity of 
. This corollary is by no means obvious from the theorem.
The fact that the form a compatible system to get a map to the inverse limit is easy because from the construction of the that we built, we can just use the same for any compatible system. From a few posts ago we worked out what the kernel of the natural surjection 
was. Call that kernel 
. To figure out the kernel of 
let’s consider the diagram:
Now notice that going around the square 
is just taking 
and sending it to the ideal we got out of this 
, but then pushing forward to 
we see that the 
is changing things by some principal and hence 
is the map that builds the ideal from alone. So we want to know the kernel and hence when is making the ideal formed from looking at 
principal.
Suppose maps to 
, then there is some 
so that 
for all . Thus 
maps to 
, the so-called unit ideles. Since this map factors through the idele class group , we haven’t changed where goes by altering it to . This shows the 
part of the diagram except for the surjection. To finish the proof of this corollary just involves messing with the ord conditions and using the Mittag-Leffler condition, so we’ll omit it. It is the statements in class field theory together with some motivating examples that are of interest for this blog at this time.
Let’s wrap up today by stating the big theorem. Suppose 
is finite abelian with conductor and 
. We get maps 
. Call the composite 
. Then
1) 
2) For all places of we get an inclusion 
by 
and 
and 
where 
is the decomposition group and 
is any place of 
over .
Now this gives us an early glimpse at why the idele formulation is going to give us more. We’ll get some control on inertia and decomposition groups.
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