The Mathieu group M23 is a Galois group over Q - X. Huang and R. Pries
Abstract
Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over \(\mathbb{Q}\) during 1984--1989.
We complete this program by proving that the last remaining sporadic group, the Mathieu group \(M_{23}\), occurs as a Galois group over \(\mathbb{Q}\). In fact, we produce an explicit degree 23 polynomial with rational coefficients whose splitting field has Galois group \(M_{23}\) over \(\mathbb{Q}\). To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of \(M_{23}\). We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of \(\mathbb{Q}(t)\) with Galois group \(M_{23}\). Essential for our computation is the numerical Belyi map algorithm developed and implemented by Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an \(M_{22}\)-extension of \(\mathbb{Q}\).
This is joint work with Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang.
- X. Huang, B. Jackson, K.-H. Lee, B. Poonen, R. Pries, S. Zhang. The Mathieu group \(M_{23}\) is a Galois group over \(\mathbb{Q}\). Preprint (2026) [ArXiV]
- M. Musty, S. Schiavone, J. Sijsling, and J. Voight, A database of Belyi maps, Proceedings of the Thirteenth Algorithmic Number Theory Symposium, Open Book Ser., vol. 2, Math. Sci. Publ., Berkeley, CA, 2019, pp. 375–392.
- G. Malle and B. H. Matzat, Inverse Galois theory, second ed., Springer Monographs in Mathematics, Springer, Berlin, 2018.