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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added the definition of idele class group to group of ideles, with a brief pointer to its role in the moduli stack of line bundles. Added both then to the function field analogy – table
added the statement of the strong approximation theorem for ideles.
added the statement of the product formula
So ideles are defined from adeles, but the way this page begins it sounds like you can have a $\mathbb{I}_{\mathbb{Z}}$, i.e., ideles from the integral adeles. And yet by definition 2, we see “The same definition holds …” and it just talks about $\mathbb{I}_{\mathbb{K}}$ for fields thereafter.
Thanks for catching this. That definition of the integral adeles has no purpose on the page, I have removed it now. Probably I (or somebody) accidentally copy-and-pasted this over when the discussion of ideles was split off from that of adeles, at some point.
I have also added some more ${}_{\mathbb{Q}}$-subscripts, for clarity.
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