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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.
Summary
Absolute values and their completions – such as 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 "Henselization". This book develops the theory of valuations as well as of Henselizations, based on the knowledge acquired in a standard graduate course in algebra.
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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)
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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)
