Xl l

INTRODUCTION

field theory and Galois theory. The generalization of finite Galois theory to infinite

normal extensions is reviewed for the reader's convenience in §13. Likewise we

develop the basic facts and formalisms of Milnor's K-theory in §§23-24 in order

not to assume any prior knowledge in this area. On the other hand, we do assume

familiarity with the language of homological algebra (exact sequences, commutative

diagrams, direct and inverse limits, etc.). The presentation is mostly self-contained,

and only very few facts are mentioned without proofs: the "snake lemma" and

some basic properties of flatness in §1.1, the structure theory of finitely generated

modules over a principal ideal domain and the Nakayama lemma in §17.4, short

cohomological discussions in §22.2, §24.3 and Remark 25.1.7, and some facts from

local class field theory in §27.1.

Unlike most existing texts on valuation theory, we chose not to develop the

theory using commutative algebra machinery, but rather to use the machinery of

abelian groups. This simplifies the presentation in many respects. The required re-

sults about abelian groups (and in particular ordered abelian groups) are developed

in Part I of the book.

Needless to say, we have not pretended to fully describe here the vast research

work done on valued and ordered fields throughout the twentieth century and which

still goes on today. The choice of material reflects only the author's personal taste

(and even more so, his limitations). More material can be found in the texts by

Ax [Ax], Bourbaki [Boul], Endler [En], Jarden [Jr], Ribenboim ([Ril], [Ri3]),

Schilling [Schi], and Zariski and Samuel [ZS] on valuation theory, as well as those by

Knebusch and Scheiderer [KnS], Lam ([Laml], [Lam2]), Prestel [Pr] and Scharlau

[Sch2] on ordered fields. Likewise, the reference list at the end of this monograph

surely covers only a small portion of the possible bibliography. Other and more

comprehensive lists of references on valuation theory can be found in [FV], [Ro],

and at the Valuation Theory internet site at http://math.usask.ca/fvk/Valth.html.

A comprehensive bibliography on the work done until 1980 on ordered fields is given

in [Laml].

I thank Eli Shamovich as well as the anonymous referees for their very valuable

comments on previous versions of this manuscript.

This book was typeset using

AMS-TJ^,

the T^X macro system of the American

Mathematical Society.

Be'er-Sheva 2005 I.E.