As usual, let be a global field. Now we do the same thing that we did last time, but for the ideles.

**1. Ideles **

First of all, we have to define the ideles. These are only a group, and are defined as the restricted direct product

relative to the unit subgroups of -units (which are defined to be if is archimedean). In other words, an idele is required to satisfy for almost all .

If is a finite set of places containing the archimedean ones, we can define the subset ; this has the product topology and is an open subgroup of . These are called the **-ideles**. As we will see, they form an extremely useful filtration on the whole idele group.

Dangerous bend: Note incidentally that while the ideles are a subset of the adeles, the induced topology on is **not** the -topology. For instance, take . Consider the sequence of ideles where is at (where is the -th prime) and 1 everywhere else. Then but not in .

However, we still do have a canonical “diagonal” embedding , since any nonzero element of is a unit almost everywhere. This is analogous to the embedding . (more…)