**Mathematical Surveys and Monographs**

Volume: 124;
2006;
288 pp;
Hardcover

MSC: Primary 12;
Secondary 19

Print ISBN: 978-0-8218-4041-2

Product Code: SURV/124

List Price: $81.00

AMS Member Price: $64.80

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-1351-4
Product Code: SURV/124.E**

List Price: $81.00

AMS Member Price: $64.80

MAA Member Price: $72.90

#### Supplemental Materials

# Valuations, Orderings, and Milnor \(K\)-Theory

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*Ido Efrat*

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.

#### Readership

Graduate students and research mathematicians interested in valuations and orderings on fields (algebra and number theory).

#### Reviews & Endorsements

This book provides a thorough and well-presented introduction to the valuations and orderings on a field, and developsa formalism of relative higher Milnor \(K\)-theory of fields.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Valuations, Orderings, and Milnor $K$-Theory

- Contents v6 free
- Introduction ix10 free
- Conventions xiii14 free
- Part I. Abelian Groups 116 free
- Part II. Valuations and Orderings 3348
- Chapter 3. Valuations 3752
- Chapter 4. Examples of Valuations 4762
- Chapter 5. Coarsenings of Valuations 5570
- Chapter 6. Orderings 6378
- Chapter 7. The Tree of Localities 6984
- Chapter 8. Topologies 7590
- Chapter 9. Complete Fields 8196
- Chapter 10. Approximation Theorems 87102
- Chapter 11. Canonical Valuations 95110
- Chapter 12. Valuations of Mixed Characteristics 107122

- Part III. Galois Theory 121136
- Chapter 13. Infinite Galois Theory 125140
- Chapter 14. Valuations in Field Extensions 127142
- Chapter 15. Decomposition Groups 133148
- Chapter 16. Ramification Theory 141156
- Chapter 17. The Fundamental Equality 151166
- Chapter 18. Hensel's Lemma 161176
- Chapter 19. Real Closures 175190
- Chapter 20. Coarsening in Algebraic Extensions 187202
- Chapter 21. Intersections of Decomposition Groups 193208
- Chapter 22. Sections 199214

- Part IV. K-Rings 205220
- Chapter 23. k-Structures 209224
- Chapter 24. Milnor K-Rings of Fields 217232
- Chapter 25. Milnor K-Rings and Orderings 225240
- Chapter 26. K-Rings and Valuations 231246
- §26.1. Valuations and extensions 231246
- §26.2. The Baer–Krull correspondence 234249
- §26.3. Totally rigid subgroups 235250
- §26.4. Sizes of multiplicative subgroups 236251
- §26.5. H[sub(s)] and the K-ring 238253
- §26.6. Bounds in the totally rigid case 240255
- §26.7. Fans 242257
- §26.8. Examples of totally rigid subgroups 244259

- Chapter 27. K-Rings of Wild Valued Fields 247262
- Chapter 28. Decompositions of K-Rings 253268
- Chapter 29. Realization of k-Structures 263278

- Bibliography 269284
- Glossary of Notation 275290
- Index 281296