p-adic integers

1 Basic construction

For any prime p, the p–adic integers is the ring obtained by taking the completion of the integers with respect to the metric induced by the norm

|x|:=1pνp(x),x, (1)

where νp(x) denotes the largest integer e such that pe divides x. The induced metric d(x,y):=|x-y| is called the p–adic metric on . The ring of p–adic integers is usually denoted by p, and its fraction field by p.

2 Profinite viewpoint

The ring p of p–adic integers can also be constructed by taking the inverse limitMathworldPlanetmath


over the inverse systemMathworldPlanetmath /p2/p0 consisting of the rings /pn, for all n0, with the projection maps defined to be the unique maps such that the diagram