Inverse Galois Problem

Inverse Galois Problem

Definitions and mathematical statement

A finite extension $L/{\mathbb{Q}}$ is Galois when it is normal and separable. Its Galois group consists of field automorphisms of $L$ fixing ${\mathbb{Q}}$. The inverse problem is \[ \forall G\text{ finite}\ \exists L/{\mathbb{Q}}\text{ finite Galois}:\qquad \operatorname{Gal}(L/{\mathbb{Q}})\simeq G. \] Only abstract group realization is demanded, not a uniform algorithm, a prescribed polynomial degree, or a regular realization over a rational-function field.

Formulation and concept references: [S1].

Short English statement

Does every finite group occur as the symmetry group of some finite Galois extension of the rationals?

Research attempt

Current frontier.

Solvable-group constructions, Hilbert irreducibility, rigidity, and embedding problems provide major realization methods, with different hypotheses [S1], Sections II--III. The August 2026 manuscript of Huang--Jackson--Lee--Poonen--Pries--Zhang states an $M_{23}$ realization with an explicit degree-$23$ polynomial [E1], Theorem 1.1 and Example 1.2. This is relevant recent work, not a theorem for every finite group; its large exact computation is not independently repeated here. The deduction below does not depend on the new realization.

Attack route.

A regular realization over ${\mathbb{Q}}(t)$ would suffice but is stronger than the target. Induction on a normal subgroup needs a proper solution of the corresponding embedding problem, not merely realization of its kernel and quotient. We instead combine a universal finite-group torsor with finitely many local Frobenius tests. The group-theoretic forcing will be proved; the global approximation step will be left visible.

Attempt: finite tests force the full group.

Fix a nontrivial finite group $G$. Choose one representative $H$ from each conjugacy class of maximal proper subgroups.

\begin{notebooklemma} There is $g_H\in G$ belonging to no conjugate of $H$. If $D\le G$ contains a conjugate of every selected $g_H$, then $D=G$. \end{notebooklemma}

Let $W={\mathbb{A}}^{|G|}_{{\mathbb{Q}}}$ be the regular permutation representation and remove the fixed loci of all nonidentity elements. The free open locus $U$ is nonempty: a vector with distinct coordinates has trivial stabilizer. Restricting the finite affine quotient gives a finite étale torsor \[ q:U\longrightarrow V=U/G. \] A rational point $v\in V({\mathbb{Q}})$ defines, up to conjugacy, a continuous homomorphism $\rho_v:G_{{\mathbb{Q}}}\to G$. Its fiber is connected precisely when $\rho_v$ is onto; a connected fiber is then a Galois field extension with group $G$. These are the standard torsor and descent conventions [E2], Sections 4.4--4.5 and 5.12.

Local construction without a global realization.

Choose distinct primes $p_H$. The unramified quotient $G_{{\mathbb{Q}}_{p_H}}\twoheadrightarrow\widehat{{\mathbb{Z}}}$ has a homomorphism to $G$ sending Frobenius to $g_H$, defining a local $G$-torsor $T_H$. Its associated étale algebra may be disconnected; that is allowed locally.

Every such $T_H$ occurs as a fiber of $q$. Twist the linear representation $W$ by $T_H$. Linear Galois descent identifies the twist with a vector space over ${\mathbb{Q}}_{p_H}$. Its twisted free locus is a nonempty Zariski-open subset and has a rational point, since an infinite field's affine space cannot be covered by finitely many proper polynomial zero sets. Untwisting gives a $G$-equivariant map $T_H\to U$ and hence a point $v_H\in V({\mathbb{Q}}_{p_H})$ with the prescribed torsor fiber.

There is an analytic neighborhood $\mathcal O_H$ of $v_H$ on which the same $G$-torsor type persists. Krasner's lemma gives local constancy of the finite étale fiber; the local identifications can be chosen equivariantly by retaining the finitely many $G$-action maps. Alternatively the twist of $q$ is an étale map with a rational point, so its image contains an analytic neighborhood and its points trivialize the twisted torsor. This uses the local openness result in [E2], Proposition 3.5.73, as well as Proposition 3.5.74.

\begin{notebookproposition} For this quotient and these neighborhoods, suppose there exists $v\in V({\mathbb{Q}})$ with \begin{equation}\label{wsref-0-72-36-31-3a-77-61}
 v_{p_H}\in\mathcal O_H\qquad\text{for every chosen }H.
\end{equation} Then $G$ occurs as a Galois group over ${\mathbb{Q}}$. Establishing this finite approximation assertion for every finite $G$ would resolve the target. \end{notebookproposition}

This asks for one specially chosen finite system of local conditions, not weak approximation for every neighborhood on every quotient. It remains a substantial arithmetic assertion, stronger than simply exhibiting a rational point of $V$.

Exact small-instance check.

For $S_3$, take $f(t)=t^3-t-1$. It is irreducible modulo $2$, being a cubic without a root. Modulo $5$, \[ f(t)=(t-2)(t^2+2t+3), \] and the quadratic has nonsquare discriminant $2$. The integer discriminant is $-23$, so the reductions are unramified. The splitting-field group contains a $3$-cycle and a transposition and is therefore $S_3$. The script verifies these elementary modular calculations. This checks the forcing mechanism, not the general approximation assertion.

Stress test: a dominant quotient map does not patch torsors.

Descending rational points of $U$ gives only trivial torsor fibers. Already for $G=C_2$, $U=\mathbb G_m$ and $q(x)=x^2$, these parameters are rational squares. They cannot enter a $p$-adic neighborhood of nonsquares. The quotient itself is rational in this example and has other rational points; the error is inferring the desired approximation from rational lifts.

The local points constructed at different primes have not been patched. Invoking Hilbert irreducibility would require a suitable rational parameter space with the needed specialization behavior, which has not been provided. Nor does the argument assume closure of realizable groups under arbitrary extensions, or infer arbitrary-group realization from realization of simple groups.

Outcome.

Outcome: Conditional reduction.

We obtain a finite maximal-subgroup obstruction set, construct its local Frobenius tests on an explicit versal quotient, and prove that one rational point meeting the tests realizes $G$. The missing step is (r61:wa) for every finite group. No general patching theorem or nonrealizable group is produced. The torsor ingredients are classical; the notebook records their assembly and exact remaining condition, without a novelty claim for the method.

Sources

• S1 Fariba Ranjbar and Saeed Ranjbar, arXiv:1512.08708 $2015$. https://arxiv.org/abs/1512.08708. Catalogue formulation source.
• S2 The Mathieu group M_23 is a Galois group over Q. https://arxiv.org/html/2608.08538v1. Catalogue research/status reference; not a proof endorsement.
• S3 The inverse Galois challenge, part II. https://galoisrepresentations.org/2026/08/16/the-inverse-galois-challenge-part-ii/. Catalogue research/status reference; not a proof endorsement.

• E1 Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang, The Mathieu group $M_{23}$ is a Galois group over ${\mathbb{Q}}$, arXiv:2608.08538 (9 August 2026), Theorem 1.1 and Example 1.2. https://arxiv.org/abs/2608.08538. Recent manuscript; its large explicit verification is not independently reproduced or used in the conditional proof. Consulted 27 September 2026.
• E2 Bjorn Poonen, Rational Points on Varieties, Graduate Studies in Mathematics 186, American Mathematical Society, 2017; Sections 4.4--4.5 and 5.12, Propositions 3.5.73--3.5.74. https://math.mit.edu/~poonen/papers/Qpoints.pdf. Descent, twists, torsors, and local étale openness/Krasner lemma. Consulted 27 September 2026.