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For any prime number , the ring of -adic integers (which, to avoid possible confusion with the ring , is also written as ) may be described in one of several ways:
To the person on the street, it may be described as (the ring of) numbers written in base , but allowing infinite expansions to the left. Thus, numbers of the form
where , added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.
More precisely, it is the metric space completion of the ring of integers with respect to the -adic absolute value. Since addition and multiplication of integers are uniformly continuous with respect to the -adic absolute value, they extend uniquely to a uniformly continuous addition and multiplication on . Thus is a topological ring.
Alternatively, it is the limit, in the category of (unital) rings, of the diagram
also considered as a topological ring if the limit is taken in the category of topological rings, and taking the rings in the diagram to have discrete topologies.
The -adic integers have the following properties:
Relation to profinite completion of the integers
The profinite completion of the integers is
This is isomorphic to the product of the -adic integers for all
(e.g. Lenstra, example 2.2)
The ring of integral adeles is the product of the profinite completion of the integers, example 1, with the real numbers
The group of units of the ring of adeles is called the group of ideles.
Introductions and surveys include
Bernard Le Stum, One century of -adic geometry – From Hensel to Berkovich and beyond talk notes, June 2012 (pdf)
Hendrik Lenstra, Profinite groups (pdf)
Revised on November 21, 2013 12:04:38
by Urs Schreiber