Adeles and Ideles
In order to understand all completions of the field ℚ of rational numbers at the same time, one forms a product of all completions, which needs to me modified in order to form a locally compact space. These modified products are discussed at length and in very general terms until they are applied to the situation at hand. This yields the ring of adeles. Its unit group is called the group of ideles. Fourier analysis on both of them is studied in quite explicit terms.
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