NIP Fields Conjecture

The conjecture proposes a complete classification of infinite fields whose first-order theory in the language of rings is NIP, that is, lacks the independence property. It predicts that any such field $K$ must be separably closed, real closed, or admit a nontrivial henselian valuation. The problem grew out of Shelah's study of dependent (NIP) theories [Shelah2005Dependent], and several equivalent or closely related formulations circulated during the 2010s, so the standard version is usually dated to around 2015.

One direction is well understood: separably closed fields are stable and real closed fields are o-minimal, so both classes are NIP, and large families of henselian valued fields are NIP as well; the work of Jahnke and Simon gives a detailed analysis of NIP henselian valued fields [JahnkeSimon2016NIP]. The conjecture asserts the converse, and it has been verified in many restricted settings, including henselian, dp-minimal, and finite-rank situations. Model-theoretic properties of general NIP fields, such as questions of decidability, have also been investigated [DittmannEtAl2023NIP].

The general case remains open: a full resolution requires showing that an arbitrary NIP field which is neither separably closed nor real closed must carry a nontrivial henselian valuation.