Inverse Galois problem: prove that M23 is a Galois group over Q.

The problem asks whether there is a finite extension of the rational numbers whose symmetry group is $M_{23}$. Huang, Jackson, Lee, Poonen, Pries, and Zhang now claim a stronger construction over $\mathbb{Q}(t)$, which specializes to the required extension over $\mathbb{Q}$.

Known results

• A 2022 account reported that no polynomial in $\mathbb{Z}[x]$ with Galois group $M_{23}$ was known; geometric realizations were available only over certain quadratic extensions.

2026 claimed construction

The preprint states that both geometric and arithmetic monodromy are $M_{23}$, proves a regular $M_{23}$-extension of $\mathbb{Q}(t)$, and gives explicit degree-$23$ polynomials whose splitting fields allegedly have group $M_{23}$. An independent account describes this as solving the problem, but the scan found no external verification or reported mathematical objection.

Current status (as of August 2026): The existence of an $M_{23}$-extension over $\mathbb{Q}$ is claimed in a new preprint, but independent confirmation is not yet recorded.