-adic integers
1 Basic construction
For any prime , the –adic integers is the ring obtained by taking the completion of the integers with respect to the metric induced by the norm
(1) |
where denotes the largest integer such that divides . The induced metric is called the –adic metric on . The ring of –adic integers is usually denoted by , and its fraction field by .
2 Profinite viewpoint
The ring of –adic integers can also be constructed by taking the inverse limit
over the inverse system consisting of the rings , for all , with the projection maps defined to be the unique maps such that the diagram