We investigate profinite structures in the sense of Newelski interpretable in fields. We show that profinite structures interpretable in separably closed fields are the same as profinite structures weakly interpretable in . We also find a strong connection with the inverse Galois problem. We give field theoretic constructions of profinite structures weakly interpretable in and satisfying some model theoretic properties, like smallness, m-normality, non-triviality, being -rank 1. For example we interpret in this way the profinite structure consisting of the profinite (...) group together with a distinguished Sylow p-subgroup of its standard structural group. (shrink)

In Dellunde et al. 997–1015), we determined the complete theory Te of modules of separably closed fields of characteristic p and imperfection degree e, eω{∞}. Here, for 0≠eω, we describe the closed set of the Ziegler spectrum corresponding to Te. Further, we establish a correspondence between certain submodules and n-types and we investigate several notions of dimensions and their relationships with the Lascar rank. Finally, we show that Te has uniform p.p. elimination of imaginaries and deduce uniform weak elimination of (...) imaginaries. (shrink)

We study the infinitely definable subgroups of the additive group in a separably closed field of finite positive imperfection degree. We give some constructions of families of such subgroups which confirm the diversity and the richness of this class of groups. We show in particular that there exists a locally modular minimal subgroup such that the division ring of its quasi-endomorphisms is not a fraction field of the ring of its definable endomorphisms, and that in contrast there exist 20 pairwise (...) orthogonal locally modular minimal subgroups whose induced structure is exactly that of an-vector space. We also show that there exist infinitely many pairwise orthogonal subgroups of infinite U-rank. Furthermore, these constructions are carried out in the additive group considered as a module over its ring of definable endomorphisms. (shrink)

RésuméPour tout entier n, on construit des sous-groupes, infiniment définissables de rang de Lascar ωn, du groupe additif d’un corps séparablement clos.