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I'm currently studying some class field theory and read about the notion of adeles and ideles. However, the object seems a bit arbitrary to me; is there a natural way to think about the adele-ring? Asked differently, why would one consider such an object in the first place?

Thanks!

Steven
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    There is an answer to a similar question here, and the answer here is certainly worth reading. See also here. – Mathmo123 Apr 21 '16 at 13:22
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    Broadly speaking, the adeles and ideles are tools that enable us to study all places of a number field at the same time, paying equal attention to each place, archimedian or not. Ideal theoretic methods tend to forget about the archimidean places (e.g. Dirichlet characters, uncompleted L-functions). Moreover, the adelic theory can be used to encode cumbersome objects (especially generalised class groups!) so that they appear more natural. It would be helpful to anyone wanting to say more if you could be a bit more specific in your question - what is it about them that seems arbitrary? – Mathmo123 Apr 21 '16 at 13:41

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