  We study the inverse Galois problem with local conditions. In particular, we
ask whether every finite group occurs as the Galois group of a Galois extension
of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian).
This property is known for all solvable groups due to Shafarevich's solution of
the inverse Galois problem for those groups. It is however completely open for
nonsolvable groups. In this paper, we provide general criteria to attack such
questions via specialization of function field extensions, and in particular
give the first infinite families of Galois realizations with only cyclic
decomposition groups and with nonsolvable Galois group. We also investigate the
analogous problem over global function fields.
