Introduction

The fundamental theorem of arithmetic describes the structure of the multi-

plicative group Q x of the field Q of rational numbers as a direct sum

Qx

^

(Z/2)

0 0 Z.

p prime

Namely, a non-zero rational number a has a unique decomposition a = ± YlpPVp(a\

where the exponents vp(a) are integers and are zero for all but finitely many primes

p. This very basic fact brings together the three main objects studied in this book:

multiplicative groups of fields, valuations, and orderings. In fact, as we shall see

later on, the maps vp are all non-trivial valuations on Q, and the ± sign corresponds

to its unique ordering.

The attempts to generalize the fundamental theorem of arithmetic to arbitrary

number fields F led to the creation of algebraic number theory. Of course, to make

such a generalization possible, one had to modify the mathematical language used.

The right generalization of both the notion of a prime number as well as of the ±

sign turned out to be that of an absolute value: a map | • | from F to the non-negative

real numbers such that \x\ = 0 if and only if x — 0, and such that

\x-y\ = \x\ • \y\ and \x + y\ \x\ + \y\

for all x,y in F. For instance, on Q the usual ordering gives an absolute value

| • |oo in the standard way, and each map vp as above gives the p-adic absolute value

\x\p =

l/pvp(x\

For the p-adic absolute value | • | = | • \p the triangle inequality can

be strengthened to the so-called ultrametric inequality

\x + y\ mix{\x\,\y\}.

Absolute values having this stronger property are called non-Archimedean, the

rest being referred to as Archimedean. Using these concepts it was possible to

develop one of the most beautiful branches of algebraic number theory: the so-

called ramification theory, which describes the behavior of absolute values under

field extensions, and especially their reflection in Galois groups.

At this point, it was natural to ask for a generalization of this theory to arbitrary

fields F. Unfortunately, the notion of an absolute value, which was satisfactory

in the number field case, is inadequate in general, so better concepts had to be

found. The right substitute for the notion of an Archimedean absolute value has

been systematically developed by E. Artin and O. Schreier in the late 1920s ([Ar],

[ASl], [AS2]), following an earlier work by Hilbert: this is the notion of an ordering

on F, i.e., an additively closed subgroup P of the multiplicative group F x of F

(standing for the set of "positive" elements) such that Fx = P U — P.

ix