Valued fields. (English) Zbl 1128.12009

Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-24221-X/hbk). x, 205 p. (2005).
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. The book is more or less self contained and gives new, interesting and short proofs of several theorems like the approximation theorem, fundamental inequality and equivalent versions of Hensels’s Lemma.
After presenting basic facts about absolute values in Chapter 1, the authors introduce valuations of arbitrary rank in Chapter 2. Chapter 3 contains study of prolongations of valuations to overfields. Chapter 4 deals with Henselian and \(p\)-Henselian fields. The structure theory for infinite Galois extensions is given in Chapter 5. The last chapter deals with the applications of general valuation theory.
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.


12J10 Valued fields
12-02 Research exposition (monographs, survey articles) pertaining to field theory
12J25 Non-Archimedean valued fields
12J15 Ordered fields
12J20 General valuation theory for fields
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