On finite embedding problems with abelian kernels

Abstract

Given a Hilbertian field k and a finite set S of Krull valuations of k, we show that every finite split embedding problem G→Gal(L/k) over k with abelian kernel has a solution Gal(F/k)→G such that every v∈S is totally split in F/L. Two applications are then given. Firstly, we solve a non-constant variant of the Beckmann–Black problem for solvable groups: given a field k and a non-trivial finite solvable group G, every Galois field extension F/k of group G is shown to occur as the specialization at some t0∈k of some Galois field extension E/k(T) of group G with E⊈k‾(T). Secondly, we contribute to inverse Galois theory over division rings, by showing that, for every division ring H and every automorphism σ of H of finite order, all finite semiabelian groups occur as Galois groups over the skew field of fractions H(T,σ) of the twisted polynomial ring H[T,σ].