Overview
- Includes new chapter on valuation rings, definable in the language of fields only
- Contains in addition a Galois theoretic characterization of the field of p-adic numbers
- Written by experts
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Monographs in Mathematics (SMM)
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Table of contents (7 chapters)
Keywords
About this book
Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization.
In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge acquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -for instance to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values alone.
Reviews
From the reviews:
"The book starts with the basic notion of absolute values followed by a comprehensive introduction to the theory of Krull valuations of arbitrary rank leading eventually to some deep results of recent research. … A useful feature of the book are its two appendices dealing with classification of V-topologies and ultraproducts of valued fields. The concise style and choice of material makes this book a wonderful reading. It is a unique, original exposition full of valuable insights." (Sudesh Kaur Khanduja, Zentralblatt MATH, Vol. 1128 (6), 2008)
Authors and Affiliations
Bibliographic Information
Book Title: Valued Fields
Authors: Antonio J. Engler, Alexander Prestel
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/3-540-30035-X
Publisher: Springer Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer-Verlag Berlin Heidelberg 2005
Hardcover ISBN: 978-3-540-24221-5Published: 13 October 2005
Softcover ISBN: 978-3-642-06345-9Published: 21 October 2010
eBook ISBN: 978-3-540-30035-9Published: 28 December 2005
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: X, 208
Topics: Algebra, Number Theory, Algebraic Geometry, Mathematical Logic and Foundations