The Mathieu group M23 is a Galois group over ℚ
Abstract
Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over ℚ during 1984-1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group M23, occurs as a Galois group over ℚ. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group M23 over ℚ. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of M23. We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of ℚ(t) with Galois group M23. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Costa, Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an M22-extension of ℚ.
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BibTeX
@misc{huang2026mathieu,
title = {{The Mathieu group $M_23$ is a Galois group over $\mathbbQ$}},
author = {Xiaoyu Huang and Blake Jackson and Kyu-Hwan Lee and Bjorn Poonen and Rachel Pries and Shaowu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.08538},
eprint = {2608.08538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08538},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Mathieu group M_23 is a Galois group over ℚ
AU - Xiaoyu Huang
AU - Blake Jackson
AU - Kyu-Hwan Lee
AU - Bjorn Poonen
AU - Rachel Pries
AU - Shaowu Zhang
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.08538
UR - https://arxiv.org/abs/2608.08538
AB - Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over 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 ℚ. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group M_23 over 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 ℚ(t) with Galois group M_23. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Costa, 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 ℚ.
ER -