2 1. Arithmetic of the p-adic Numbers

1.1. From Q to R; the concept of completion

The real numbers, denoted by R, are obtained from the rationals by

a procedure called completion. This procedure can be applied to any

metric space, i.e., to a space M with a metric d on it. Recall that a

function

d: M x M -*m

defined on all ordered pairs (a?, y) of elements of a nonempty set M

is said to be a metric if it possesses the following properties:

(1) d(x, y) 0; d(x, y) = 0 if and only if x = y;

(2) d(xiy) = d(y}x) y x.yeM;

(3) d(x, y) d(x} z) + d(z, y)V x,y,ze M.

The function d is also called the distance function.

We say that a sequence {rn} in a metric space M is a Cauchy

sequence if for any e 0 there exists a positive integer N such that

n,m N implies d(rn, rm) e. If any Cauchy sequence in M has a

limit in M, then M is called a complete metric space.

Theorem 1.1 (Completion Theorem). Every metric space M can be

completed, i.e., there exists a metric space (M,D) such that

(1) M is complete with respect to the metric D;

(2) M contains a subset Mo isometric to M;

(3) Mo is dense in M (i.e., each point in M is a limit point for

M0).

The proof that can be found e.g. in [13, Theorem 76] consists

in an explicit construction of the completion M and the metric D

on it. We start with the collection {M} of all Cauchy sequences in

My convergent or not, and turn it into a metric space. But first we

introduce an equivalence relation on {M}: two Cauchy sequences an

and bn are called equivalent, we write {an} ~ {6n}, if d(anibn) — 0.

(It is easy to check that this is an equivalence relation on {M}.)

We define M to be the set of equivalence classes, M = {M}/ ~.

The metric D between two equivalence classes of Cauchy sequences