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 use a non-rigid triple of conjugacy classes of $M_{23}$ and compute Belyi maps 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.