Basic Number Theory pp 59-79 | Cite as

# Adeles

Chapter

## Abstract

Throughout this Chapter,
where the second product is taken over all the places of

*k*will denote an**A**-field; if*v*is a place of*k*,*k*_{ v }will denote the completion of*k*at*v*; if*v*is a finite place of*k*, we write*r*_{ v }for the maximal compact subring of*k*_{ v }and*p*_{ v }for the maximal ideal of*r*_{ v }, these being the subsets of*k*_{ v }respectively defined by |x|_{ v }< 1 and by |x|_{ v }< 1. We write*P*_{ ∞ }for the set of the infinite places of*k*, and*P*for any finite set of places of*k*, containing*P*_{ ∞ }. For any such set*P*, put$$ {k_A}(P) = \prod\limits_{v \in P} {{k_v}} \times \prod\limits_{v \notin P} {{r_v}} , $$

(1)

*k*, not in*P*. With the usual product topology, this is locally compact, since the*k*_{ v }are so and the*r*_{ v }are compact. On*k*_{ A }(*P*), we put a ring structure by defining addition and multiplication componentwise; clearly this makes*k*_{ A }(*P*) into a topological ring. Set-theoretically,*k*_{ A }(*P*) could be defined as the subset of the product π*k*_{ v }consisting of the elements*x =*(*x*_{ v }) of that product such that |x_{ v }|v < 1 for all*v*not in*P.*If*P′*is also a finite set of places of*k*, and*P′**P*, then*k*_{ A }(*P*) is contained in*k*_{ A }(*P′*); moreover, its topology and its ring structure are those induced by those of*k*_{ A }(*P′*) and*k*_{ A }(*P*) is an open subset of*k*_{ A }(*P′*).## Keywords

Discrete Subgroup Finite Dimension Open Subgroup Topological Ring Canonical Injection
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1973