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Asymptotic Differential Algebra and Model Theory of Transseries$
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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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Differential-Henselian Fields

Differential-Henselian Fields

(p.340) Chapter Seven Differential-Henselian Fields
Asymptotic Differential Algebra and Model Theory of Transseries

Matthias Aschenbrenner

Lou van den Dries

Joris van der Hoeven

Princeton University Press

This chapter discusses differential-henselian fields. Here K is a valued differential field with small derivation. An extension of K means a valued differential field extension of K whose derivation is small. After some preliminaries about d-henselianity, the chapter proves Theorem 7.0.1 stating that if k is linearly surjective and K is d-algebraically maximal, then K is d-henselian. For monotone K with linearly surjective k it proves the uniqueness-up-to-isomorphism-over-K of maximal immediate extensions. It also considers the case of few constants and shows that in the presence of monotonicity (perhaps unnecessary) a converse to Theorem 7.0.1 can be obtained. Finally, it describes differential-henselianity in several variables.

Keywords:   differential-henselian field, valued differential field, small derivation, d-henselianity, d-henselian, maximal immediate extension, constant, monotonicity, differential-henselianity

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