More advanced remarks on \(\widehat{\mathbb{Z}}\) versus \(\mathbb{Q}\)

This section can be skipped on first reading.

People who have seen some more advanced algebra might recognise this construction of \(\widehat{\mathbb{Z}}\) as being the profinite completion of the additive abelian group \(\mathbb{Z}\), so it is a fundamental object of mathematics in some sense. But usually, when building mathematics, after \(\mathbb{Z}\) we go to \(\mathbb{Q}\), a multiplicative localisation of \(\mathbb{Z}\), and only complete after that (to get \(\mathbb{R}\)). The process of “completing before localising” gives us a far more arithmetic completion of \(\mathbb{Z}\).

Even though \(\mathbb{Q}\) is a divisible abelian group and hence its profinite completion vanishes, we can still attempt to "locally profinitely complete it" by defining \(\widehat{\mathbb{Q}}:=\mathbb{Q}\otimes_{\mathbb{Z}}\widehat{\mathbb{Z}}\). This object is more commonly known as the finite adeles of \(\mathbb{Q}\). More generally if \(F\) is any number field then \(F\otimes_{\mathbb{Z}}\widehat{\mathbb{Z}}\) is the ring of finite adeles of \(F\). To get to the full ring of adeles of a number field \(F\) you need to take the product with the ring of infinite adeles of \(F\), which is \(F\otimes_{\mathbb{Q}}\mathbb{R}\): some kind of universal archimedean completion of \(F\). I don't know a reference which develops the theory of adeles in this way, so this is what we shall do here.

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