Every sporadic simple group is a Galois group over Q

The inverse Galois problem asks whether every finite group occurs as the Galois group of a finite Galois extension of Q. The problem has a long and interesting history, starting with Gauss in 1801. Among the finite simple groups, the 26 sporadic groups have received special attention. During 1984--1989, the rigidity method and its braid-theoretic refinements realized 25 of the 26 sporadic groups as Galois groups over Q. The exception was the Mathieu group M23, which resisted every attempt for nearly four decades. In our recent paper, we prove that M23 occurs as a Galois group over Q, and in fact as the Galois group of a regular extension of Q(t), meaning it occurs infinitely often. I will describe the history, the construction, the verification, and the role that AI tools played in this project that began at the May 2026 AIM workshop on AI and Number Theory.

This is joint work with Xiaoyu Huang, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang.