Strongly dependent ordered abelian groups have finite dp-rank. They are precisely those groups with finite spines and |{p prime : [G : pG] = ∞}| < ∞. We apply this to show that if K is a strongly dependent field, then (K, v) is strongly dependent for any henselian valuation v. 1 2. Preliminaries and notation Throughout the text G will denote a group, usually abelian and often ordered, C will denote a sufficiently saturated model of Th(G). By definable we will mean definable with parameters. We will need a few results from [30]. Since this text is not readily available, we try to keep the present work as self contained as possible, referring to more accessible sources whenever we are aware of such. In particular, for the study of ordered abelian groups we chose the language of [5], rather than the language used by Schmitt. The next sub-section is dedicated to a quick overview of (parts) of the language we are using, and to the basic properties of definable sets.2.1. Ordered abelian groups. Recall that an abelian group (G; +) is orderd if it is equipped with a linear ordering < such that a < b implies a + g < b + g for all a, b, g ∈ G. An ordered abelian group is discrete if it has a minimal positive element, and dense otherwise. It is archimedean if for all a, b ∈ G there exists n ∈ Z such that

Generalising Hrushovski's fusion technique we construct the free fusion of two strongly minimal theories T1. T2 intersecting in a totally categorical sub-theory T0. We show that if. e.g., T0 is the theory of infinite vector spaces over a finite field then the fusion theory Tω, exists, is complete and ω-stable of rank ω. We give a detailed geometrical analysis of Tω, proving that if both T1, T2 are 1-based then. Tω can be collapsed into a strongly minimal theory, if some additional technical conditions hold—all trivially satisfied if T0 is the theory of infinite vector spaces over a finite field .

We prove elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if (K, v) is strongly dependent then so is its henselization.1 See [22, Appandix A] for a more detailed discussion.

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We survey the history of Shelah's conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.unpublished work with R. Cluckers.