On finite embedding problems with abelian kernels

Given a Hilbertian field k and finite set S of Krull valuations , we show that every split embedding problem G → Gal ( L / ) over with abelian kernel has solution F such v ∈ is totally in . Two applications are then given. Firstly, solve non-constant variant the Beckmann–Black for solvable groups: given non-trivial group Galois extension shown to occur as specialization at some t 0 E T ⊈ ‾ Secondly, contribute inverse theory division rings, by showing that, ring H automorphism σ order, all semiabelian groups skew fractions twisted polynomial [ ]