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)

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

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see (...) text] where [Formula: see text] is a saturated separably closed field. (shrink)