
Contents
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Notes and comments Notes and comments
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Here is a key consequence of our assumption ∂O ⊆ O: LEMMA 6.1.1. If y 2 O, then (y0)n+1 4 yn for all n. Here is a key consequence of our assumption ∂O ⊆ O: LEMMA 6.1.1. If y 2 O, then (y0)n+1 4 yn for all n.
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Notes and comments Notes and comments
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6.2 ALGEBRAIC EXTENSIONS 6.2 ALGEBRAIC EXTENSIONS
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Unramified and purely ramified algebraic extensions Unramified and purely ramified algebraic extensions
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Preserving monotonicity Preserving monotonicity
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Notes and comments Notes and comments
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6.4 The Valuation Induced on the Value Group 6.4 The Valuation Induced on the Value Group
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A more precise estimate on vP A more precise estimate on vP
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Notes and comments Notes and comments
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6.5 Asymptotic Couples 6.5 Asymptotic Couples
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Notes and comments Notes and comments
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Lemmas on equalizing Lemmas on equalizing
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Notes and comments Notes and comments
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6.8 EVALUATION AT PSEUDOCAUCHY SEQUENCES 6.8 EVALUATION AT PSEUDOCAUCHY SEQUENCES
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6.9 CONSTRUCTING CANONICAL IMMEDIATE EXTENSIONS 6.9 CONSTRUCTING CANONICAL IMMEDIATE EXTENSIONS
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Six Valued Differential Fields
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Published:June 2017
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Abstract
This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ → Γ for homogeneous P ∈ K K{Y}superscript Not Equal To. The chapter then shows that the derivation of any valued differential field extension of K that is algebraic over K is also small. It also explains how differential field extensions of the residue field k give rise to valued differential field extensions of K with small derivation and the same value group. Finally, it discusses asymptotic couples, dominant part, the Equalizer Theorem, pseudocauchy sequences, and the construction of canonical immediate extensions.
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