We study the algebraic implications of the non-independence property (NIP) and variants thereof (dp-minimality) on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a (definable) henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson's preprint "dp-minimal fields", arXiv:to find the right analogue of super-stability in the context of NIP theories, Shelah introduced the notion of strong NIP. As part of establishing this analogy, Shelah showed [32, Claim 5.40] that the theory of a separably closed field that is not algebraically closed is not strongly NIP. In fact Shelah's proof actually shows that strongly NIP fields are perfect 1 . Shelah conjectured [32, Conjecture 5.34] that (interpreting its somewhat vague formulation) strongly NIP fields are real closed, algebraically closed or support a definable non-trivial (henselian) valuation. Recently, this conjecture was proved 2 by Johnson [20] in the special case of dp-minimal fields (and, independently, assuming the definability of the valuation, henselianity is proved in [19]).The two main open problems in the field are:(1) Let K be an infinite (strongly) NIP field that is neither separably closed nor real closed. Does K support a non-trivial definable valuation? (2) Are all (strongly) NIP fields henselian (i.e., admit some non-trivial henselian valuation) or, at least, t-henselian (i.e., elementarily equivalent in the language of rings, to a henselian field)?A positive answer to Questions (2) would imply, for example, that strongly 3 dependent fields are elementarily equivalent to Hahn fields over well understood base fields [11, Theorem 3.11]:Equi-characteristic: R((t Γ )), C((t Γ )) or F p ((t Γ )). Finite residue field: Q((t Γ )) where Q is a p-adically closed field if the field admits a henselian valuation with finite residue field. Kaplansky: L((t Γ )) where L is a rank 1 Kaplansky field with residue field as in (1) above.where in all cases Γ is a strongly dependent ordered abelian group (see [10] for the classification of such groups).In view of the above, a natural strategy for studying Shelah's conjecture would be to, on the one hand, study the conjecture for Hahn fields (with dependent residue fields), as the key example and -on the other handusing the information gained in the study of Hahn fields, try to generalise Johnson's results from dp-minimal fields to the strongly dependent setting.The simplest extension of Johnson's proof of Shelah's conjecture for dpminimal fields would be to finite extensions of dp-minimal fields. Section 2 is dedicated to showing that this extension is vacuous, namely we prove that a finite extension of a dp-minimal field is again dp-minimal (see Theorem 1 Shelah's proof only uses the simple fact that if char(K) = p > 0 then either K is perfect or [K × : (K × ) p ] is infinite. See, e.g., [27, Remark 2.5] 2

In [Koe94] Koenigsmann shows that every field that admits a t-henselian topology is either real closed or separably closed or admits a definable valuation inducing the t-henselian topology. To show this Koenigsmann investigates valuation rings induced by certain (definable) subgroups of the field. The aim of this paper, based on the authors PhD thesis [Dup15], is to look at the methods used in [Koe94] in greater detail and correct a mistake in the original paper based on [JK15a]. * key words and phrases: valuations; definable valuations; q-henselian valued fields; t-henselian topologies MSC classes: 03C40 03C60 12J10 12L12 1.3 Lemma. Let O be a valuation ring and A an O-ideal. Then (1 + A) is a multiplicative subgroup of O × .Proof: It is clear that 1 + A ⊆ O × . Let a ∈ A. Then 1 + a ∈ O × . Hence (1 + a) −1 ∈ O × . Therefore a · (1 + a) −1 ∈ A and hence (1 + a) −1 = 1 + a · (1 + a) −1 ∈ 1 + A. Further for a, b ∈ A we have (1 + a) · (1 + b) = 1 + a + b + a · b ∈ 1 + A.

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