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
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
the T^X macro system of the American
Be'er-Sheva 2005 I.E.