The inverse Galois problem over formal power series fields

Introduction The inverse Galois problem asks whether every finite group G occurs as a Galois group over the field Q of rational numbers. We then say that G is realizable over Q. This problem goes back to Hilbert [Hil] who realized Sn and An over Q. Many more groups have been realized over Q since 1892. For example, Shafarevich [Sha] finished in 1958 the work started by Scholz 1936 [Slz] and Reichardt 1937 [Rei] and realized all solvable groups over Q. The last ten years have seen intensified efforts toward a positive solution of the problem. The area has become one of the frontiers of arithmetic geometry (see surveys of Matzat [Mat] and Serre [Se1]). Parallel to the effort of realizing groups over Q, people have generalized the inverse Galois problem to other fields with good arithmetical properties. The most distinguished field where the problem has an affirmative solution is C(t). This is a consequence of the Riemann Existence Theorem from complex analysis. Winfried Scharlau and Wulf-Dieter Geyer asked what is the absolute Galois group of the field of formal power series F = K((X1, . . . , Xr)) in r ≥ 2 variables over an arbitrary field K. The full answer to this question is still out of reach. However, a theorem of Harbater (Proposition 1.1a) asserts that each Galois group is realizable over the field of rational function F (T ). By a theorem of Weissauer (Proposition 3.1), F is Hilbertian. So, G is realizable over F . Thus, the inverse Galois problem has an affirmative solution over F . The goal of this note is to prove the same result in a more general setting.