x INTRODUCTION

The proper definition in the non-Archimedean case is more subtle, and was

introduced by W. Krull in his landmark 1931 paper [Kru2]. Roughly speaking,

instead of looking at the absolute value | • | itself, Krull focused on the group

homomorphism v — — log | • |:

Fx

— » R. Of course, this minor modification cannot

change much, and is still insufficient for general fields. However, Krull's conceptual

breakthrough was to replace the additive group R by an arbitrary ordered abelian

group (T, ). Thus what we now call a Krull valuation on the field F is a group

homomorphism v: Fx — » T, where (T, ) is an ordered abelian group, which satisfies

the following variant of the ultrametric inequality:

v(x + y) min{i;(x), v(y)}

for x 7^ —y.

Krull's seminal work [Kru2] paved the way to modern valuation theory. Start-

ing from this definition, he introduced some of the other key ingredients of the

theory: valuation rings, the analysis of their ideals, the convex subgroups of (T, ),

and the connections between all these objects and coarsenings of valuations. He

adapted for his general setting the (already existent) notions of decomposition, in-

ertia, and ramification subgroups of Galois groups over F. Furthermore, he studied

maximality properties of valued fields with respect to field extensions. In a some-

what more implicit way he also studied a notion which will later on become central

in valuation theory, namely, Henselian valued fields (although he does not give it

a name). This notion turned out to be the right algebraic substitute in the setup

of Krull valuations for the topological property of completeness. It is analogous to

the notion of a real closed Geld introduced by Artin and Schreier in the context of

ordered fields. The term "Henselian" is in honor of K. Hensel, who discovered the

field Qp of p-adic numbers, and proved (of course, under a different terminology)

that its canonical valuation is Henselian [He]. We refer to [Ro] for a comprehensive

study of the early (pre-Krull) history of valuation theory.

The classical theory of valuations from the point of view of Krull and his follow-

ers is well presented in the already classical books by O. Endler [En], P. Ribenboim

[Ril], and O.F.G. Schilling [Schi]. Yet, over the decades that elapsed since the

publication of these books, valuation theory went through several conceptual de-

velopments, which we have tried to present in this monograph.

First, the different definitions in the Archimedean and non-Archimedean cases

caused a split of the unified theory into two separate branches of field arithmetic:

the theory of ordered fields on one hand, and valuation theory on the other hand.

While Krull still keeps in [Kru2] a relatively unified approach (at least to the extent

possible), later expositions on general valuation theory have somewhat abandoned

the connections with orderings. Fortunately, the intensive work done starting in the

1970s on ordered fields and quadratic forms (which later evolved into real algebraic

geometry) revived the interest in this connection, and led to a reintegration of

these two sub-theories. T.Y. Lam's book [Lam2] beautifully describes this interplay

between orderings and valuations from the more restrictive viewpoint of the reduced

theory of quadratic forms, i.e., quadratic forms modulo a preordering (see also

[Laml] and [Jr]). In the present book we adopt this approach in general, and

whenever possible study orderings and valuations jointly, under the common name

localities.

Second, starting already from Krull's paper [Kru2], the emphasis in valuation

theory has been on its Galois-theoretic aspects. These will be discussed in detail in