Regular version of the inverse Galois problem for skew fields

Abstract

In order to extend the study of the regular version of the regular inverse Galois problem to skew fields, we generalize the definition of regular field extensions for commutative fields to the case of arbitrary fields. We then propose a general version of the inverse Galois property and show that for a field k, the study of the class InvReg(k) of non-trivial regular extensions of k satisfying this property constitutes a natural generalization of the classical regular inverse Galois problem. Next, we use recent results from Behajaina, Deschamps, and Legrand on the inverse Galois problem in the noncommutative setting to show that certain skew fraction fields h(t, {\sigma}, {\delta}) belong to the class InvReg(h). Finally, we generalize Behajaina's method to construct extensions belonging to InvReg(h) that are not of the form h(t, {\sigma}) with {\sigma} an automorphism of finite order.