A central conjecture of inverse Galois theory, formulated by Dèbes and Deschamps in 1997, asserts that for every field kk and every finite split embedding problem over the rational function field k(T)k(T) — that is, an epimorphism α:G→Aut⁡(L/k(T))\alpha:G\to\operatorname{Aut}(L/k(T)) with a group-theoretic section, where L/k(T)L/k(T) is a finite Galois extension and GG a finite group — there exists a regular Galois solution: a finite Galois extension E/k(T)E/k(T) containing LL with E∩kˉ=L∩kˉE\cap\bar{k}=L\cap\bar{k} and an isomorphism β:Aut⁡(E/k(T))→G\beta:\operatorname{Aut}(E/k(T))\to G compatible with α\alpha and the restriction map. The conjecture is known for ample fields (Pop, Haran–Jarden) and for separably closed fields. Fehm, Legrand, and Paran (2019) proved unconditionally that regular solutions always exist once the normality of E/k(T)E/k(T) is waived, isolating normality of the solution field as precisely the open core. The full conjecture remains open: it implies the Regular Inverse Galois Problem over every field, in particular over Q\mathbb{Q}, which is still the ultimate open goal of the theory as of 2026, and no counterexample is known.

Background

The classical inverse Galois problem (IGP), open since the late 19th century, asks whether every finite group GG occurs as the Galois group of a Galois extension of Q\mathbb{Q}. A geometric strengthening is the Regular Inverse Galois Problem (RIGP): realize GG as the Galois group of a finite Galois extension E/Q(T)E/\mathbb{Q}(T) that is regular over Q\mathbb{Q}, meaning E∩Q‾=QE\cap\overline{\mathbb{Q}}=\mathbb{Q}, so that no new constants are introduced. By Hilbert's irreducibility theorem — valid over Hilbertian fields, i.e., fields for which that theorem holds, such as all global fields — a regular realization over Q(T)\mathbb{Q}(T) specializes to a realization over Q\mathbb{Q}, so RIGP implies IGP.

The natural framework subsuming these questions is that of finite embedding problems. A finite embedding problem for a field KK consists of a finite group GG, a finite Galois extension L/KL/K with group Aut⁡(L/K)\operatorname{Aut}(L/K), and an epimorphism α:G→Aut⁡(L/K)\alpha:G\to\operatorname{Aut}(L/K); the kernel of α\alpha is the subgroup ker⁡α⊆G\ker\alpha\subseteq G. The problem is called split if α\alpha admits a group-theoretic section (a homomorphic embedding Aut⁡(L/K)→G\operatorname{Aut}(L/K)\to G with α∘α′=id⁡\alpha\circ\alpha'=\operatorname{id}). A proper (Galois) solution is a finite Galois extension F/KF/K with L⊆FL\subseteq F together with an isomorphism β:Aut⁡(F/K)→G\beta:\operatorname{Aut}(F/K)\to G such that α∘β\alpha\circ\beta equals the restriction map Aut⁡(F/K)→Aut⁡(L/K)\operatorname{Aut}(F/K)\to\operatorname{Aut}(L/K). Realization questions are special cases: taking the trivial quotient recovers the problem of realizing GG itself, and a solution field whose intersection with the algebraic closure of the base is unchanged is called regular. When K=k(T)K=k(T) is a rational function field, a regular solution is one with E∩k‾=L∩k‾E\cap\overline{k}=L\cap\overline{k}.

In this language, Dèbes and Deschamps (1997) formulated a central conjecture of inverse Galois theory: every finite split embedding problem over k(T)k(T) has a regular Galois solution, for an arbitrary base field kk. The conjecture implies the RIGP over every field, and hence the IGP over every Hilbertian field, including Q\mathbb{Q}. It is proved when kk is an ample field (also called large: every smooth curve over kk with one kk-rational point has infinitely many), by Pop (1996) via patching, and by Haran–Jarden (1998); related solvability over Hilbertian PAC fields goes back to Fried–Völklein (1992). A PAC (pseudo-algebraically closed) field is one over which every geometrically irreducible variety has a dense set of rational points. Koenigsmann (2004) constructed the first field over which the RIGP is solvable although it contains no ample field, showing the class of good fields is strictly larger than the ample ones, without settling the conjecture there.

A separate, weaker lineage asks only that every finite group occur as the automorphism group (not necessarily Galois group) of a finite extension of Q\mathbb{Q}: Fried–Kollár (1978) and Fried (1980) settled this, and Legrand–Paran (2018) extended it to Hilbertian fields. Fehm–Legrand–Paran (2019) unified these threads: they proved unconditionally that every finite split embedding problem over k(T)k(T), for arbitrary kk, admits a regular solution E/k(T)E/k(T) with the required group isomorphism, except that E/k(T)E/k(T) need not be normal (Galois). Thus the entire content of the Dèbes–Deschamps conjecture that remains open is the normality of the solution field. The standard group-theoretic route — deducing solvability of all embedding problems from projectivity of the absolute Galois group (a profinite group is projective if every finite embedding problem for it is solvable) — is obstructed in general: as Fehm–Legrand–Paran record in Remark 3.4, for a PAC field kk of characteristic zero that is not algebraically closed, the absolute Galois group of k(T)k(T) is not projective, so the class of fields over which all Galois embedding problems over k(T)k(T) are solvable is genuinely not understood beyond the ample and separably closed cases.

Problem Statement

Settle the Dèbes–Deschamps conjecture in its full Galois form, as stated as Conjecture 1.1 by Fehm–Legrand–Paran (following Dèbes–Deschamps 1997, §2.2):

Let kk be an arbitrary field, GG a finite group, L/k(T)L/k(T) a finite Galois extension of the rational function field, and α:G→Aut⁡(L/k(T))\alpha:G\to\operatorname{Aut}(L/k(T)) an epimorphism admitting a group-theoretic section (a finite split embedding problem). Must there exist a finite Galois extension E/k(T)E/k(T) with L⊆EL\subseteq E and E∩k‾=L∩k‾E\cap\overline{k}=L\cap\overline{k} (regularity of the solution), together with an isomorphism β:Aut⁡(E/k(T))→G\beta:\operatorname{Aut}(E/k(T))\to G such that α∘β\alpha\circ\beta is the restriction map Aut⁡(E/k(T))→Aut⁡(L/k(T))\operatorname{Aut}(E/k(T))\to\operatorname{Aut}(L/k(T))?

The common scientific objective is to determine the status of regular Galois solvability of all finite split embedding problems over k(T)k(T) for arbitrary base fields; the conjecture is settled by a proof for every field kk, or by an explicit field kk together with a finite split embedding problem over k(T)k(T) that admits no regular Galois solution. The known positive cases are ample fields (by Pop's theorem) and separably closed fields; the class of base fields for which every finite split embedding problem over k(T)k(T) has a regular Galois solution is not understood beyond these cases. No restriction on kk, on the finite group, or on the split embedding problem may be assumed; the conjecture is known for arbitrary kk only with the normality of E/k(T)E/k(T) waived (Fehm–Legrand–Paran 2019).

References

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