Embedding problems for automorphism groups of field extensions

Abstract

A central conjecture in inverse Galois theory, proposed by Dèbes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this conjecture, namely that such embedding problems can be regularly solved if one waives the requirement that the solution fields are normal. This extends previous results of M. Fried, Takahashi, Deschamps and the last two authors concerning the realization of finite groups as automorphism groups of field extensions.

$Θ$-Hilbertianity and strong $Θ$-Hilbertianity

“…By [FLP,Proposition 3.3] the constant embedding problem (res K(t)s/L : Gal(K(t)) → A, α : B → A) has a solution. Thus, there exists a Galois extension N of K(t) containing L and an isomorphism θ : Gal(N/K(t)) → B such that α • θ = res N/L .…”

“…Finally, let us recall the PRC counterpart of [FLP,Proposition 3.3]: Lemma 3.3. [FHV,Theorem 5.2] Let K be a PRC field.…”

On a variant of the Beckmann--Black problem

“…Function field extensions with specified specializations. A generalization of [DL21, théorème A], in the direction of finite embedding problems, is studied by Fehm, Paran, and the author in [FLP19]. In §3 of the present paper, we also go further than producing finite separable field extensions E/k(T ) with E ∩ k = k and specified automorphism groups, but in another direction: we construct such extensions with specified specializations.…”