- Title Pages
- Conventions and Notations
- Dramatis Personæ
- Introduction and Overview
- Chapter One Some Commutative Algebra
- Chapter Two Valued Abelian Groups
- Chapter Three Valued Fields
- Chapter Four Differential Polynomials
- Chapter Five Linear Differential Polynomials
- Chapter Six Valued Differential Fields
- Chapter Seven Differential-Henselian Fields
- Chapter Eight Differential-Henselian Fields with Many Constants
- Chapter Nine Asymptotic Fields and Asymptotic Couples
- Chapter Ten <i>H</i>-Fields
- Chapter Eleven Eventual Quantities, Immediate Extensions, and Special Cuts
- Chapter Twelve Triangular Automorphisms
- Chapter Thirteen The Newton Polynomial
- Chapter Fourteen Newtonian Differential Fields
- Chapter Fifteen Newtonianity of Directed Unions
- Chapter Sixteen Quantifier Elimination
- Appendix A Transseries
- Appendix B Basic Model Theory
- List of Symbols
- (p.110) Chapter Three Valued Fields
- Asymptotic Differential Algebra and Model Theory of Transseries
Lou van den Dries
Joris van der Hoeven
- Princeton University Press
This chapter introduces the reader to basic field theory by focusing on valued fields. It first considers valuations on fields before discussing the basic properties of valued fields, with emphasis on extensions. It then describes pseudoconvergence in valued fields, along with henselian valued fields. It also shows how to decompose a valuation on a field into simpler ones, leading to an analysis of various special types of pseudocauchy sequences. Because the valuation of is compatible with its natural ordering, some basic facts about fields with compatible ordering and valuation are presented. The chapter concludes by reviewing some basic model theory of valued fields as well as the Newton diagram and Newton tree of a polynomial over a valued field.
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