Open problem, Algebra, posed 1892

Inverse Galois problem

Open

Does every finite group $G$ occur as the Galois group $\operatorname{Gal}(K/\mathbb{Q})$ of some finite Galois extension $K$ of the field of rational numbers $\mathbb{Q}$ ?

Research frontier as of 2026

As of 2026, the general Inverse Galois Problem over $\mathbb{Q}$ remains open. Every solvable group is realized (Shafarevich 1954), and an August 2026 preprint by Huang, Jackson, Lee, Poonen, Pries, and Zhang completed the sporadic program by realizing the last holdout, the Mathieu group $M_{23}$ , via a non-rigid triple of conjugacy classes and numerical Belyi map computations. However, infinite families of finite simple groups of Lie type (such as $\operatorname{PSL}_n(\mathbb{F}_q)$ for arbitrary $n$ and $q$ ) and general nonsolvable group extensions remain out of reach.

Best known results

Every finite solvable group is a Galois group over $\mathbb{Q}$ (Shafarevich 1954).
All 26 sporadic finite simple groups are realized over $\mathbb{Q}$ : 25 via rigidity methods (Thompson, Matzat, Malle, Pahlings, and others, 1984–1989) and $M_{23}$ in a 2026 preprint by Huang, Jackson, Lee, Poonen, Pries, and Zhang.

Tools and where they stop

Tool	Achieved	Where it stops
Hilbert's irreducibility theorem and Thompson's rigidity method	Constructs regular Galois extensions of $\mathbb{Q}(t)$ from rigid tuples of rational conjugacy classes in $G$ , realizing $S_n$ , $A_n$ , 25 sporadic groups, and many classical groups of Lie type	Many simple groups of Lie type over arbitrary finite fields $\mathbb{F}_{p^r}$ lack rational rigid tuples, and rigidity does not automatically lift through arbitrary group extensions
Cohomological embedding problems and patched Galois representations	Solves the Inverse Galois Problem for all solvable groups (Shafarevich) and realizes linear groups $\operatorname{PSL}_2(\mathbb{F}_{p^r})$ for restricted congruence classes via modular forms and automorphic representations	Brauer–Manin obstructions and lack of general automorphic lifting theorems prevent uniform realization of all non-abelian extensions and higher-rank groups of Lie type

Open questions

Can every finite simple group — in particular $\operatorname{PSL}_2(\mathbb{F}_{p^r})$ for all primes $p$ and exponents $r \ge 1$ — be realized as a regular Galois group over $\mathbb{Q}(t)$ ?

References

Jean-Pierre Serre (1992). Topics in Galois Theory
Gunter Malle, B. Heinrich Matzat (1999). Inverse Galois Theory · DOI:10.1007/978-3-662-12123-8
Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, Shaowu Zhang (2026). The Mathieu group M23 is a Galois group over Q · arXiv:2608.08538 [preprint, not peer-reviewed]