Mathematical Surveys and Monographs 124. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4041-X/hbk). xiii, 288 p. $ 75.00 (2006).
The monograph under review is a systematic and almost entirely self-contained exposition of the general theory of valued and ordered fields. It reflects the conceptual and technical development of the research in this area carried out for the last 30-35 years. The main objects dealt with are multiplicative groups of fields, valuations and orderings (brought together by various relations discussed in the text). It is well-known that the origin of the theory and the initial motivation for its development came from the role of absolute values of global fields in algebraic number theory and Diophantine analysis. Note also that the creation of abstract algebra in the 1920s, and the role of the Artin-Schreier theory of ordered fields in solving Hilbert’s 17-th Problem, stimulated the efforts to extend the results on absolute values from global to arbitrary fields. As explained in the Introduction, the implementation of this program led to the modern point of view on the topics discussed in the book; it caused, however, a split of the unified theory into two separate branches, focusing on ordered fields and on Krull valuations, respectively. Unlike many valuation-theorists, the author does not neglect the connections between valuations and orderings but emphasizes them, and whenever possible, studies them jointly, under the common name localities. He prefers to build up the theory by applying the machinery of ordered abelian groups instead of traditionally used techniques of commutative algebra. The book presents the classical aspects of the considered fields, namely, their arithmetic, topology and Galois theory. The reader can find such key ingredients of the theory as: valuation rings, the analysis of their ideals, the convex subgroups of the value group, the connection between these objects and coarsenings of valuations. The weak approximation theorem is stated in a general form characterizing independent localities; incomparable valuations are characterized in a similar manner. Apart from this, completions play a limited role in the general theory (except the case of real valued valuations). The properties of valuation prolongations on algebraic (particularly, finite) extensions are presented comprehensively. For a normal extension $L$ of a valued field $(F, v)$, it gives information on decomposition, inertia and ramification subgroups of the Galois group $G(L/F)$. By showing when a valuation $v$ of $F$ is relatively Henselian with respect to $L$, the author introduces the reader to the important classes of Henselian valued fields (and more generally, of $p$-Henselian valued fields for a prime number $p$, solvably Henselian valued fields, e.t.c.). He includes important characterizations of relative Henselity like Hensel’s lemma, the Hensel-Rychlik condition and the Krasner-Ostrowski lemma. Relative real closures of ordered fields are viewed as analogues to relative Henselizations; they are characterized in the framework of a relative Artin-Schreier theory (by applying Sturm’s theorem for real closed fields). The concluding quarter of the book is devoted to cohomological aspects of valuations and orderings. They are discussed using the language of Milnor’s $K$-theory (more precisely, the introduced generalized version of the Milnor $K$-ring functor, and its target category, the so-called $k$-structures). The discussion introduces the reader to valuation-theoretic techniques as used in applications to birational abelian geometry. Throughout the monography, the presented material is illustrated by examples and constructions. This quality of the book (as well as its bibliography) could make it genuinely useful for graduate students in algebra and number theory, and for mathematicians of different levels of interest in valuation theory and its applications.