Model-theoretic combinatorial complexity and algebraic properties are interlinked in somewhat surprising ways. An example of that is a well-known result by Kaplan, Scanlon and Wagner, which states that infinite NIP fields of characteristic p have no Artin-Schreier extension. This result has since then being proven to also hold for n-dependent fields, and a weaker version holds for NTP2 fields. In all these settings, we can reduce the argument down to an explicit formula, and use it to lift complexity from the residue field; most notably, this allows us to obtain a classification of n-dependent henselian valued fields.