Mathematicians closed a 40-year gap in Galois theory in three months -- using AI as "a telescope," not a solver

In August, six mathematicians -- Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries and Shaowu Zhang -- closed a gap in Galois theory that had sat open since the 1980s. Their paper (arXiv:2608.08538), posted on Aug. 9, proves that the Mathieu group M23 occurs as a Galois group over the rational numbers -- the last of the 26 "sporadic" simple groups to be realized this way, after mathematicians nailed down the other 25 between 1984 and 1989. The team didn't just prove existence -- they built the object itself: an explicit degree-23 polynomial with rational coefficients whose splitting field has Galois group M23 over Q, using computed Belyi maps to construct the extension directly.

The inverse Galois problem asks a deceptively simple question: given a group of symmetries, can you find a polynomial whose roots have exactly that symmetry structure? For most groups, yes -- but the 26 sporadic groups are exceptions to every known pattern in the classification of finite simple groups, which made each one its own standalone puzzle rather than an instance of a general method. M23, one of the smallest sporadic groups, turned out to be the hardest of the 26, resisting repeated attempts across four decades. The project that finally cracked it took about 3 months from start to finish, beginning at a Caltech workshop in late May, where the American Institute of Mathematics had gathered mathematicians specifically to identify problems suited to AI-assisted computation -- Rachel Pries brought M23 to that room as a candidate.

That same workshop season produced a second, parallel result on a related but distinct question. Fields Medalist Terence Tao's Foundation for Science and AI Research ran a crowdsourced competition -- open to "unrestricted use of any sort of computational tool, including AI" -- asking teams to find polynomials for all 25,000 transitive permutation groups acting on 24 points. That challenge concluded in late August with every one of the 25,000 signatures found, won by a German team that Yahoo's reporting says used AI only for a minor script-writing task, not for the mathematics itself. The two efforts ran on overlapping timelines and the same underlying problem family, but they are not the same project: Tao's competition breadth-searched a huge grid of easier cases with a scoring system that rewarded difficulty, while the Huang-Jackson-Lee-Poonen-Pries-Zhang paper went deep on the single hardest holdout.

What AI actually did inside the M23 project is the part worth reading past the headline. The team used Claude Fable 5, Claude Opus 4.8 and GPT-5.6 Sol -- generating and testing code, organizing large computations, working with existing mathematical software packages, and monitoring and recovering from failed runs. Kyu-Hwan Lee's own description is unglamorous and specific: "We could do it very efficiently. That wasn't really possible five years ago." At one point, an AI agent testing a coordinate transformation reported back "this might work" -- and, per the team's account, it did; Scientific American's reporting describes what followed as feeling "miraculous" to the mathematicians watching it unfold.

Co-author Shaowu Zhang wrote his own account of the AI's role afterward, and it reads as a deliberate correction against overstatement rather than a victory lap. "We also tried more autonomous approaches at the beginning of the project," Zhang writes. "Within our limited AI-usage budget, these attempts did not make meaningful progress." The decisive moves, in his telling, were entirely human: "deciding what to compute, stopping an increasingly expensive calculation, changing the strategies, interpreting the numerical evidence, and deciding where to direct our limited resources." No text in the manuscript itself was written by AI.

“AI gave us a powerful and precise telescope, but the team's mathematical knowledge told us where to point it, when further magnification would not help and how to interpret what came into view.” — Shaowu Zhang, co-author, on Proofs and Prompts

Nothing about the M23 result names a specific next target -- with all 26 sporadic groups now realized, that particular open list is closed. But Tao's SAIR competition has a second stage planned, explicitly aimed at the harder, more conceptual cases the first stage's breadth-first scoring system was designed to skip over, with no date yet announced. (The three models used were all already general-purpose assistants, not math-specialized systems -- the team's account credits the workflow and human direction, not a purpose-built research tool, which is part of why Zhang's telescope framing has landed with other mathematicians rather than being dismissed as lab-specific hype.) What actually traveled out of this project is the division of labor itself: AI for the code, the search, the failure recovery and the grunt computation; humans for every decision about where to look and when to stop looking. That template has no expiration date tied to M23 being solved, which is exactly why the mathematicians who ran it are the same ones cautioning against reading it as a bigger result than it is.