Contents
Introduction ix
Conventions xiii
Part I. Abelian Groups
Chapter 1. Preliminaries on Abelian Groups 5
§1.1. General facts 5
§1.2. Divisible hulls 7
§1.3. Rational ranks 11
§1.4. Characters 12
Chapter 2. Ordered Abelian Groups 15
§2.1. Basic properties and examples 15
§2.2. Ranks 17
§2.3. Cores 19
§2.4. Cofinality and infinitesimals 20
§2.5. Ordered abelian groups of rank 1 21
§2.6. Push-downs 23
§2.7. Well-ordered sets 24
§2.8. Formal power series 27
§2.9. Generalized rational functions 30
Part II. Valuations and Orderings
Chapter 3. Valuations 37
§3.1. Valuation rings 37
§3.2. Valuations 38
§3.3. Places 42
§3.4. Discrete valuations 43
Chapter 4. Examples of Valuations 47
§4.1. Valuations from unique factorization domains 47
§4.2. Valuations on power series fields 48
§4.3. Gauss valuations 50
Chapter 5. Coarsenings of Valuations 55
§5.1. Coarser and finer 55
§5.2. Quotients and compositions of valuations 56
§5.3. Coarsenings in the mixed characteristic case 60
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