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 t-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.
Will Johnson (Fri,) studied this question.