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
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