- Title Pages
- Epigraph
- Preface
- Conventions and Notations
- Leitfaden
- 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 H-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 - Bibliography
- List of Symbols
- Index
Valued Fields
Valued Fields
- Chapter:
- (p.110) Chapter Three Valued Fields
- Source:
- Asymptotic Differential Algebra and Model Theory of Transseries
- Author(s):
Matthias Aschenbrenner
Lou van den Dries
Joris van der Hoeven
- Publisher:
- 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.
Keywords: valued field, valuation, extension, pseudoconvergence, henselian valued field, pseudocauchy sequence, Newton diagram, Newton tree
Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.
Please, subscribe or login to access full text content.
If you think you should have access to this title, please contact your librarian.
To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.
- Title Pages
- Epigraph
- Preface
- Conventions and Notations
- Leitfaden
- 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 H-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 - Bibliography
- List of Symbols
- Index
