Probl\`emes de plongement finis sur les corps non commutatifs
Abstract
We extend finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of a skew field H of finite dimension over its center h. First, we show that solving a finite embedding problem over H is equivalent to finding a solution to some finite embedding problem over h fulfilling a polynomial constraint. Next, we show that every constant finite split embedding problem over the skew field of fractions H(t) with central indeterminate t has a solution, if h is an ample field. This is a non-commutative analogue of a deep result of Pop. More generally, we solve such finite embedding problems over the skew field of fractions H(t, σ) of the twisted polynomial ring H[t, σ], for some automorphisms σ of H of finite order. Our results extend previous works on the inverse Galois problem over skew fields.