Problèmes de plongement finis sur les corps non commutatifs

Abstract

We extend the notion of finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of centrally finite division rings. We first show that solving a finite embedding problem over such a division ring H is equivalent to finding a solution to some finite embedding problem over the center of H involving a polynomial constraint. We then show that every constant finite split embedding problem over the skew field of rational fractions H(t) with central indeterminate t has a solution, if the center of H is an ample field, thus providing with a non-commutative analogue of a deep result of Pop. More generally, we solve such finite embedding problems over skew fields of rational fractions H(t, σ), where σ denotes an arbitrary automorphism of H with finite order. Our results generalize previous works on the inverse Galois problem over skew fields.