The inverse Galois problem (over number fields k) is one of the central problems in algebraic number theory. A classical approach to it is via specialization of Galois coverings: Hilbert’s irreducibility theorem guarantees the existence of infinitely many specialization values in k such that the Galois group of the specialization equals the Galois group of the covering. I will consider problems related to the inverse Galois problem which can be attacked using the specialization approach.
In particular, the Grunwald problem is a strengthening of the inverse Galois problem, asking about the existence of Galois extensions with prescribed Galois group which approximates finitely many prescribed local extensions. I will explain some of the ideas and difficulties behind solving Grunwald problems via the specialization approach. I will also present some new observations about the structure of the set of all specializations of a Galois covering.
Parts of this talk are based on joint work with Francois Legrand and Danny
Neftin.