First, an example of a 2‐dependent group without a minimal subgroup of bounded index is given. Second, all infinite n‐dependent fields are shown to be Artin‐Schreier closed. Furthermore, the theory of any non separably closed PAC field has the IPn property for all natural numbers n and certain properties of dependent (NIP) valued fields extend to the n‐dependent context.

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Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.2000 Mathematics Subject Classification. 03C45, 03C60, 12E15.