Valuations, Orderings, and Milnor K-Theory

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Overview

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 them in 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 introduction to 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.

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Product Details

  • ISBN-13: 9780821840412
  • Publisher: American Mathematical Society
  • Publication date: 2/28/2006
  • Series: Mathematical Surveys and Monographs , #124
  • Pages: 288
  • Product dimensions: 7.20 (w) x 10.30 (h) x 0.90 (d)

Table of Contents

Ch. 1 Preliminaries on Abelian groups 5
Ch. 2 Ordered Abelian groups 15
Ch. 3 Valuations 37
Ch. 4 Examples of valuations 47
Ch. 5 Coarsenings of valuations 55
Ch. 6 Orderings 63
Ch. 7 The tree of localities 69
Ch. 8 Topologies 75
Ch. 9 Complete fields 81
Ch. 10 Approximation theorems 87
Ch. 11 Canonical valuations 95
Ch. 12 Valuations of mixed characteristics 107
Ch. 13 Infinite Galois theory 125
Ch. 14 Valuations in field extensions 127
Ch. 15 Decomposition groups 133
Ch. 16 Ramification theory 141
Ch. 17 The fundamental equality 151
Ch. 18 Hensel's lemma 161
Ch. 19 Real closures 175
Ch. 20 Coarsening in algebraic extensions 187
Ch. 21 Intersections of decomposition groups 193
Ch. 22 Sections 199
Ch. 23 K-structures 209
Ch. 24 Milnor K-rings of fields 217
Ch. 25 Milnor K-rings and orderings 225
Ch. 26 K-rings and valuations 231
Ch. 27 K-rings of wild valued fields 247
Ch. 28 Decompositions of K-rings 253
Ch. 29 Realization of k-structures 263
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