On a variant of the Beckmann--Black problem

Abstract

Given a field k and a finite group G, the Beckmann-Black problem asks whether every Galois field extension F/k with group G is the specialization at some t 0 ∈ k of some Galois field extension E/k(T ) with group G and E ∩ k = k. We show that the answer is positive for arbitrary k and G, if one waives the requirement that E/k(T ) is normal. In fact, our result holds if Gal(F/k) is any given subgroup H of G and, in the special case H = G, we provide a similar conclusion even if F/k is not normal. We next derive that, given a division ring H and an automorphism σ of H of finite order, all finite groups occur as automorphism groups over the skew field of fractions H(T, σ) of the twisted polynomial ring H[T, σ].