# Valued Fields

###### A. J. Engler and A. Prestel
Publisher:
Springer Verlag
Publication Date:
2005
Number of Pages:
205
Format:
Hardcover
Series:
Springer Monographs in Mathematics
Price:
89.95
ISBN:
3-540-24221-X
Category:
Monograph
[Reviewed by
Álvaro Lozano-Robledo
, on
05/25/2006
]
This book is a comprehensive introduction to the general theoryof valuations on fields. Let us recall the most general definition of a general valuation, which is due to Krull. Let G be an ordered abelian group and let ∞ be a symbol denoting infinity. A valuation on a field K is a surjective map v from K to G ∪ {∞} satisfying, for all x and y in K, the following axioms:
1. if v(x)=∞ then x=0,
2. v(xy) = v(x) + v(y), and
3. v(x+y) ≥ min{v(x),v(y)}.

Notice that if G=R is the additive group of real numbers, and we drop the surjectivity condition, then by defining |x|=e–v(x) we obtain a classical non-archimedean absolute value, i.e. it satisfies the ultrametric triangle inequality |x+y| ≤ max{|x|,|y|}. The general theory of valuations is of great interest, in particular, in number theory. For example, the valuations and completions of the rational numbers (and number fields) lead to useful and interesting ideas like the p-adic numbers, local fields, Galois cohomology arguments, etc.

Engler and Prestel describe in detail the theory of valuations, starting from the basics and gradually building up (at a very reasonable pace). The first chapter is devoted to absolute values and completions of fields with respect to fixed absolute values. The second chapter defines valuations in general, shows how to construct valuations and describes the topology induced by a valuation. The third chapter concentrates on answering the following question: when and in how many ways can a valuation v on K be extended to a valuation on a field L containing K (Chevalley's extension theorem and so on).

A field K complete with respect to a (classical) non-archimedean absolute value satisfies the well-known Hensel's lemma, an extremely useful result. However, a field K with a more general valuation (with an arbitrary group G as above) may not satisfy Hensel's lemma. However, there is an algebraic extension of K, which we call the Henselization of K, where Hensel's lemma holds once again. Chapters 4 and 5 of Valued Fields  are devoted to the study of Henselizations and Henselian fields. Chapter 5 explains the structure theory of Henselian fields and the connections with the Galois theory of certain infinite extensions. Finally, chapter 6 treats applications of valuation theory: Artin's conjecture, p-adically closed fields and a local-global principle for quadratic forms. The book also contains two appendices, on ultraproducts of valued fields and the classification of V-topologies.

As hinted above, the book is written quite nicely and makes for some enjoyable reading (if your interests bring you to the theory of valuations, of course). The volume is essentially self-contained: the authors cover most of the material that is needed or they briefly summarize the most important results (e.g., infinite Galois theory). Perhaps, the first few chapters could be optimized. For example, the definitions of valuation, ring of integers and the maximal ideal appear in the introduction, in the first chapter and then again in the second chapter, where the most general definition is finally given. But once again, the pace and exposition of concepts in the text seemed, overall, of excellent quality.

Finally, I must say that the price of the book (\$89.95 for 200 pages) seems hard to justify (regardless of the content), as it happens with many other volumes by the same publisher.

Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1 Absolute Values – Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Archimedean Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Non-Archimedean Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Ordered Abelian Groups – Valuations . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Constructions of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.1 Rational Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Ordered Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.3 Rigid Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Dependent Valuations – Induced Topology . . . . . . . . . . . . . . . . . . 42

2.4 Approximation – Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Extension of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1 Chevalley’s Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Algebraic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 The Fundamental Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Transcendental Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1 Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 p-Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3 Ordered Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4 The Canonical Henselian Valuation . . . . . . . . . . . . . . . . . . . . . . . . 103

4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

X Contents

5 Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.1 Infinite Galois Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Unramified Extensions – First Exact Sequence . . . . . . . . . . . . . . 120

5.3 Ramified Extensions – Second Exact Sequence . . . . . . . . . . . . . . 126

5.4 Galois Characterization of Henselian Fields . . . . . . . . . . . . . . . . . 136

5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Applications of Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.1 Artin’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.2 p-Adically Closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.3 A Local-Global Principle for Quadratic Forms . . . . . . . . . . . . . . . 163

A Ultraproducts of Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

B Classification of V -Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Standard Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203