• André Weil
Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 144)

## Abstract

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