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Asymptotic Differential Algebra and Model Theory of Transseries$
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Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven

Print publication date: 2017

Print ISBN-13: 9780691175423

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691175423.001.0001

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PRINTED FROM PRINCETON SCHOLARSHIP ONLINE (www.princeton.universitypressscholarship.com). (c) Copyright Princeton University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in PRSO for personal use.date: 25 February 2021

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
DOI:10.23943/princeton/9780691175423.003.0004

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

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