We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as of a extension Q all whose decomposition groups are cyclic (resp., abelian). This property is known for solvable due to Shafarevich's solution those groups. It however completely open nonsolvable this paper, provide general criteria attack such questions via specialization funct...

This paper surveys the methods that have been used to attack conjecture, still open, an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.