See attached file.

The English Wikipedia page on p-adic numbers explains them well and has references to literature for studying them (e.g. Gouvea's book). Their construction is given by the standard way of completing the rational numbers with respect to an absolute value --- but this time that absolute value is the p-adic norm. The Archimedian Axiom is violated, because the metric induced by the p-adic norm is an ultrametric: it satisfies the strict triangle inequality

d(x,y)

This is a very good question. In addition to the helpful observations already made, there is a bit more to add.

A good introduction to p-adic integers is given in Section 3.1 in the attached paper by Semmes in the attached arXiv preprint.

Feldman gives an introduction to the group of p-adic numbers in the attached arXiv preprint. The main theorem is given in Section 2.