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Valuation theory is used constantly in algebraic number theory and field theory, and is currently gaining considerable research interest. Ribenboim fills a unique niche in the literature as he presents one of the first introductions to classical valuation theory in this up-to-date rendering of the authors long-standing experience with the applications of the theory. The presentation is fully up-to-date and will serve as a valuable resource for students and mathematicians.
These are the revised notes of a course for graduate students and some seminar talks which I gave at the University of Rochester during Fall Term 1969/70. They would not have been written without the encouragement and the aid which I received, during all stages of the work, by friends from Rochester, Rio de Janeiro, and Bonn. I wish to thank all of them: Barbara Grabkowicz encouraged me to write these notes in English and read carefully parts of a preliminary manuscript, as did Gervasio G. Bastos, Yves A. E. Lequain, Walter Strubel, and Antonio J. Engler. Many valuable suggestions were given me by Yves A. E. Lequain, and several improvements of theorems and proofs are due to him. I am particularly grateful to Linda C. Hill for her criticism in reading the last version and for improving and smoothing many of my formulations. Last but not least I thank Wilson Goes for the excellent typing. Most of this book was elaborated when I stayed in Rio de Janeiro as a Visiting Professor at IMPA (Institute for Pure and Applied Mathematics) and as a Pesquisador-Conferencista of CNPq (National Research Council). Thanks are also due to these institu tions.
For more than a century, valuation theory has had its classical roots in algebraic number theory, algebraic geometry and the theory of ordered fields and groups. In recent decades it has seen an amazing expansion into many other areas. Moreover, having been dormant for a while in algebraic geometry, it has now been reintroduced as a tool to attack the open problem of resolution of singularities in positive characteristic and to analyze the structure of singularities. Driven by this topic, and by its many new applications in other areas, the research in valuation theory itself has also been intensified, with a particular emphasis on the deep open problems in positive characteristic. The multifaceted development of valuation theory has been monitored by two International Conferences and Workshops: the first in 1999 in Saskatoon, Canada, and the second in 2011 in Segovia and El Escorial in Spain. This book grew out of the second conference and presents high quality papers on recent research together with survey papers that illustrate the state of the art in several areas and applications of valuation theory. This book is addressed to researchers and graduate students who work in valuation theory or the areas where it is applied, as well as a general mathematical audience interested in the expansion and usefulness of the valuation theoretical approach, which has been called the ``most analytic'' form of algebraic reasoning. For young mathematicians who want to enter these areas of research, it provides a valuable source of up-to-date information.
Classical valuation theory has applications in number theory and class field theory as well as in algebraic geometry, e.g. in a divisor theory for curves. But the noncommutative equivalent is mainly applied to finite dimensional skewfields. Recently however, new types of algebras have become popular in modern algebra; Weyl algebras, deformed and quantized algebras, quantum groups and Hopf algebras, etc. The advantage of valuation theory in the commutative case is that it allows effective calculations, bringing the arithmetical properties of the ground field into the picture. This arithmetical nature is also present in the theory of maximal orders in central simple algebras. Firstly, we aim at uniting maximal orders, valuation rings, Dubrovin valuations, etc. in a common theory, the theory of primes of algebras. Secondly, we establish possible applications of the noncommutative arithmetics to interesting classes of algebras, including the extension of central valuations to nice classes of quantized algebras, the development of a theory of Hopf valuations on Hopf algebras and quantum groups, noncommutative valuations on the Weyl field and interesting rings of invariants and valuations of Gauss extensions.
Theory of valuations on convex sets is a classical part of convex geometry which goes back at least to the positive solution of the third Hilbert problem by M. Dehn in 1900. Since then the theory has undergone a multifaceted development. The author discusses some of Hadwiger's results on valuations on convex compact sets that are continuous in the Hausdorff metric. The book also discusses the Klain-Schneider theorem as well as the proof of McMullen's conjecture, which led subsequently to many further applications and advances in the theory. The last section gives an overview of more recent developments in the theory of translation-invariant continuous valuations, some of which turn out to be useful in integral geometry. This book grew out of lectures that were given in August 2015 at Kent State University in the framework of the NSF CBMS conference “Introduction to the Theory of Valuations on Convex Sets”. Only a basic background in general convexity is assumed.
This book is the first of two proceedings volumes stemming from the International Conference and Workshop on Valuation Theory held at the University of Saskatchewan (Saskatoon, SK, Canada). Valuation theory arose in the early part of the twentieth century in connection with number theory and has many important applications to geometry and analysis: the classical application to the study of algebraic curves and to Dedekind and Prufer domains; the close connection to the famousresolution of the singularities problem; the study of the absolute Galois group of a field; the connection between ordering, valuations, and quadratic forms over a formally real field; the application to real algebraic geometry; the study of noncommutative rings; etc. The special feature of this book isits focus on current applications of valuation theory to this broad range of topics. Also included is a paper on the history of valuation theory. The book is suitable for graduate students and research mathematicians working in algebra, algebraic geometry, number theory, and mathematical logic.
This book is the second of two proceedings volumes stemming from the International Conference and Workshop on Valuation Theory held at the University of Saskatchewan (Saskatoon, SK, Canada). It contains the most recent applications of valuation theory to a broad range of mathematical ideas. Valuation theory arose in the early part of the twentieth century in connection with number theory and continues to have many important applications to algebra, geometry, and analysis. The research and survey papers in this volume cover a variety of topics, including Galois theory, the Grunwald-Wang Theorem, algebraic geometry, resolution of singularities, curves over Prufer domains, model theory of valued fields and the Frobenius, Hardy fields, Hensel's Lemma, fixed point theorems, and computations in valued fields. It is suitable for graduate students and research mathematicians interested in algebra, algebraic geometry, number theory, and mathematical logic.
This monograph is a comprehensive exposition of the modern theory of valued and ordered fields. It presents the classical aspects of such fields: their arithmetic, topology, and Galois theory. Deeper cohomological aspects are studied in its last part in an elementary manner. This is done by means of the newly developed theory of generalized Milnor $K$-rings. The book emphasizes the close connections and interplay between valuations and orderings, and to a large extent, studies themin a unified manner. The presentation is almost entirely self-contained. In particular, the text develops the needed machinery of ordered abelian groups. This is then used throughout the text to replace the more classical techniques of commutative algebra. Likewise, the book provides an introductionto the Milnor $K$-theory. The reader is introduced to the valuation-theoretic techniques as used in modern Galois theory, especially in applications to birational anabelian geometry, where one needs to detect valuations from their ``cohomological footprints''. These powerful techniques are presented here for the first time in a unified and elementary way.