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.
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