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

The problem asks whether the Mathieu group $M_{23}$ occurs as the Galois group of a finite extension of $\mathbb{Q}$. Before 2026, it was the only sporadic simple group without a known realization over $\mathbb{Q}$.

Known results

• In 2022, realizations over fields including $\mathbb{Q}(\sqrt{-23})$ and $\mathbb{Q}(\sqrt{-7})$ were known, but no polynomial over $\mathbb{Q}$ with Galois group $M_{23}$ was known.
• By 1989, the other $25$ sporadic simple groups had been realized over $\mathbb{Q}$; $M_{23}$ remained exceptional.

August 11, 2026 claimed construction

Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang claim a regular $M_{23}$-extension of $\mathbb{Q}(t)$, an explicit degree-$23$ polynomial over $\mathbb{Q}$, and examples unramified outside $\{2,3,23\}$. The preprint presents this as a proof, with Magma and PARI/GP certification, but it remains unrefereed and independently unverified.

Current status (as of September 2026): A $2026$ preprint claims an $M_{23}$-extension over $\mathbb{Q}$, but independent verification is not recorded.