Chapter 1

Arithmetic of the p-adic

Numbers

The aim of the first chapter of this book is to introduce its main

protagonist: the field of p-adic numbers Qp, defined for any prime p.

Just like the field of real numbers R, the field Qp can be con-

structed from the rational numbers Q as its completion with respect

to a certain norm. This norm depends on the prime number p and

differs drastically from the standard Euclidean norm used to define

R. Nevertheless, in each of the two cases, completion yields a normed

field (R and Qp), and this general concept is studied in detail in §1.2.

But first (§1.1), we recall the completion procedure in the more fa-

miliar case of the reals (this takes us from Q to R), and only then do

we go on to its generalization to arbitrary normed fields (§1.3).

Putting these preliminaries aside, we come to the central section

of Chapter 1 (§1.4), where the construction of Qp is actually carried

out.

§§1.5-1.8 are devoted to the algebraic and structural properties

of the p-adic numbers. Here, as in subsequent parts, we will be con-

stantly comparing Qp and R, stressing both their similarities and their

differences. Finally, §§1.9 and 1.10 treat additional topics and are not

closely related to the rest of the book.

1

http://dx.doi.org/10.1090/stml/037/01