2021/12/22 by François Legrand, Legrand, François
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Commutative Algebra and Its Applications #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Polynomial and algebraic computation #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2112.12170
openalex publication_date 2021/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Hilbertian field k and a finite set S of Krull valuations of k, we show that every finite split embedding problem G → \rmGal(L/k) over k with abelian kernel has a solu\-tion \rmGal(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 \not ⊆ 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].