Arithmetic & Homotopic Galois Theory IRN | The Mathieu group M23 is a Galois group over Q

September 14, 2026	The Mathieu group M23 is a Galois group over Q Rachel Pries and Xiaoyu Huang, Colorado State University, US and Temple University, US RIMS Kyoto (Room 110) + Zoom · JP: 21:00 · FR: 14:00 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.
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September 14, 2026

The Mathieu group M23 is a Galois group over Q

Rachel Pries and Xiaoyu Huang, Colorado State University, US and Temple University, US RIMS Kyoto (Room 110) + Zoom · JP: 21:00 · FR: 14:00

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.