The Mathieu group $M_{23}$ is a Galois group over $\mathbb{Q}$

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 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 $\mathbb{Q}$.

Lots of additional information, 22 pages. Simpler equation for the regular M_23-extension of Q(t). More info about its specializations (bonus: a new realization of M_22 as a Galois group over Q). More info about coefficient recognition and rigorous Galois group computation. More info about the other six regular M_23-extensions over a degree 6 field arising from this Nielsen class