- Title Pages
- Conventions and Notations
- Dramatis Personæ
- Introduction and Overview
Chapter OneSome Commutative Algebra
Chapter TwoValued Abelian Groups
Chapter ThreeValued Fields
Chapter FourDifferential Polynomials
Chapter FiveLinear Differential Polynomials
Chapter SixValued Differential Fields
Chapter SevenDifferential-Henselian Fields
Chapter EightDifferential-Henselian Fields with Many Constants
Chapter NineAsymptotic Fields and Asymptotic Couples
Chapter ElevenEventual Quantities, Immediate Extensions, and Special Cuts
Chapter TwelveTriangular Automorphisms
Chapter ThirteenThe Newton Polynomial
Chapter FourteenNewtonian Differential Fields
Chapter FifteenNewtonianity of Directed Unions
Chapter SixteenQuantifier Elimination
Appendix BBasic 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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