vi CONTENTS
Chapter 6. Orderings 63
§6.1. Ordered fields 63
§6.2. Examples of orderings 66
§6.3. Archimedean orderings 67
Chapter 7. The Tree of Localities 69
§7.1. Localities 69
§7.2. Localities on residue fields 70
§7.3. The tree structure 71
Chapter 8. Topologies 75
§8.1. Basic properties 75
§8.2. Continuity of roots 77
§8.3. Bounded sets 79
Chapter 9. Complete Fields 81
§9.1. Metrics 81
§9.2. Examples 82
§9.3. Completions 83
Chapter 10. Approximation Theorems 87
§10.1. Approximation by independent localities 87
§10.2. Approximation by incomparable valuations 90
§10.3. Consequences 93
Chapter 11. Canonical Valuations 95
§11.1. Compatible localities 95
§11.2. S-cores 98
§11.3. Explicit constructions 100
§11.4. Existence of valuations 103
Chapter 12. Valuations of Mixed Characteristics 107
§12.1. Multiplicative representatives 107
§12.2. A-adic expansions 109
§12.3. p-perfect structures 110
§12.4. Rings of Witt vectors 116
§12.5. Mixed valuations under a finiteness assumption 118
Part III. Galois Theory
Chapter 13. Infinite Galois Theory 125
Chapter 14. Valuations in Field Extensions 127
§14.1. Chevalley's theorem 127
§14.2. Valuations in algebraic extensions 128
§14.3. The Galois action 130
Chapter 15. Decomposition Groups 133
§15.1. Definition and basic properties 133
§15.2. Immediateness of decomposition fields 134
§15.3. Relatively Henselian fields 136
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