Mathematicians solve the last sporadic Galois case with AI help

A six-author paper posted to arXiv on August 9 settles a question that had stood since the 1980s: the Mathieu group M23, the last of the 26 sporadic simple groups without a known polynomial, is a Galois group over the rational numbers. Scientific American reported on Tuesday, September 22, how the team got there, and the account includes three months of work by humans and AI agents.

The acknowledgments say so plainly. The authors write that they used “Claude Fable 5, Claude Opus 4.8, and ChatGPT 5.6 Sol” for searching the literature, generating code, testing hypotheses, ruling out approaches, devising computational strategies, checking results and proofreading. They add: “No text in this article was written by AI.”

What was actually proved

The inverse Galois problem asks whether every finite group can appear as the symmetry group of the roots of some polynomial with rational coefficients. The sporadic groups are the 26 finite simple groups that fit no infinite family, and in the 1980s mathematicians found polynomials for 25 of them. M23 was the exception. “There’s this one last holdout,” Bjorn Poonen of MIT told Scientific American.

The paper, by Xiaoyu Huang of Temple University, Blake Jackson of Carnegie Mellon’s Institute for Computer-Aided Reasoning in Mathematics, Kyu-Hwan Lee of the University of Connecticut, Poonen, Rachel Pries of Colorado State and Shaowu Zhang of Caltech, produces an explicit degree-23 polynomial with rational coefficients whose splitting field has Galois group M23. The construction runs through Belyi maps, and the authors describe a numerical Belyi-map algorithm by Klug, Musty, Schiavone, Sijsling and Voight as essential to the computation. One specialisation of the extension also yields an M22 extension of the rationals.

Where the machines came in

Pries presented the problem in May at an American Institute of Mathematics conference at Caltech that had asked for questions suited to AI’s capacity for large-scale search. Within three months the group had a solution. According to the magazine’s account, AI searched M23 for usable combinations of symmetries and numerically approximated equations for seven candidate surfaces; when the approximations refused to resolve into exact numbers even at 90 digits, the humans realised the coordinate system was the problem, and a batch of agents was set to try new coordinates. Most failed. One came back with a workable surface and the message “This might work!”

“We could do it very efficiently,” Lee said. “That wasn’t really possible five years ago.” Pries: “This problem had been open for a very, very long time.”

Epoch AI, which had listed the M23 case among its FrontierMath open problems, first marked it as a human solution and then reclassified it as “human + AI” after the authors said AI had been instrumental. That reclassification is the part to hold on to. Nobody is claiming a language model proved a theorem. The claim is that models did enough of the searching, coding and checking that the humans consider the result a joint one.

The counter-example in the same story

Scientific American also describes a crowdsourced competition, run by the Foundation for Science and AI Research, to find polynomials for all 25,000 relationships between polynomials and symmetry groups acting on 24 roots. The first phase closed in late August with every group accounted for. The winners were a pair of German mathematicians who used AI only to write an upload script. “It was open to AI, and it was still these folks who did the best,” said Jen Paulhus of Mount Holyoke College, one of the organisers.

Put the two results side by side and you get a fair picture of where AI in pure mathematics stands in September 2026. On a problem that turned on a human insight about coordinates, agents did the grinding search that made the insight pay. On a broad hunt across thousands of cases, trained people still beat everyone.

What is not settled

The paper is a preprint. It was presented at a seminar of the Arithmetic and Homotopic Galois Theory network on September 14, but it has not yet appeared in a peer-reviewed journal, and the magazine’s narrative of how the agents worked is the authors’ account rather than an independent one. Watch for the referees’ verdict, and for whether the same pairing of numerical algorithms and agents can crack any of the other open problems Epoch keeps on its list.