Translating between NIP integral domains and topological fields

In this paper, we prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application, we prove several results about definable ring topologies on NIP fields, including the following. Let [Formula: see text] be an NIP field or expansion of a field. Let [Formula: see text] be a definable ring topology on [Formula: see text]. Then [Formula: see text] is a field topology, and [Formula: see text] is locally bounded. If [Formula: see text] has characteristic [Formula: see text] or finite dp-rank, then [Formula: see text] is “generalized [Formula: see text]-henselian” in the sense of Dittman, Walsberg and Ye, meaning that the implicit function theorem holds for polynomials. If [Formula: see text] has finite dp-rank, then [Formula: see text] must be a topology of “finite breadth” (a [Formula: see text]-topology). Using these techniques, we give some reformulations of the conjecture that NIP local rings are henselian.