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The restricted product of all completions of a global field with respect to their valuation rings at almost all non-Archimedean places.
wp-adele-ringmilne-algebraic-number-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
The adeles combine all local completions with an integrality restriction at almost all finite places.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
milne-algebraic-number-theoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
The restricted-product topology makes the adeles a locally compact topological ring for a global field.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
milne-algebraic-number-theorystacks-top-ringtopology-without-tears