On a variant of the Beckmann--Black problem

2021/11/13 by Legrand, François
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2111.07155

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 t₀ ∈ 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 \rmGal(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].

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