The inverse Galois problem for the Mathieu group M23

Huang, Jackson, Lee, Poonen, Pries and Zhang gave an explicit degree-23 polynomial over ℚ with Galois group M23, the last sporadic simple group not yet realized; Epoch AI credits AI as instrumental, but the paper has not been peer reviewed.

Problem
Is there a polynomial over Q\mathbb{Q} whose Galois group is the Mathieu group M23M_{23}?
Credit
Huang, Jackson, Lee, Poonen, Pries and Zhang, with Claude and ChatGPT models · Human-AI collaboration; Belyi-map computations
Checked
Marked solved (human + AI) by Epoch AI
Timeline
arXiv 2608.08538 v1 2026-08-09, v2 2026-10-07
Sources
Epoch AI problem pagePaper (arXiv:2608.08538)
UN-
VERIFIEDHuman-led · AI role per Epoch AI

The problem

The inverse Galois problem asks whether every finite group occurs as the Galois group of a polynomial over Q\mathbb{Q}. Of the 26 sporadic simple groups, the Mathieu group M23M_{23} was the last for which no such polynomial was known.

What AI did

Huang, Jackson, Lee, Poonen, Pries and Zhang posted a paper on 9 August 2026 (arXiv:2608.08538) giving an explicit polynomial of degree 23 with Galois group M23M_{23}, found through computations with Belyi maps. Epoch AI credits Claude Fable 5, Claude Opus 4.8 and ChatGPT 5.6 Sol as instrumental, but the arXiv abstract page does not mention AI.

How it was checked

The second version of the paper (7 October 2026) mentions a rigorous computation of the Galois group. The paper has not been peer reviewed, and we found no independent check.

Limitations

Epoch AI first marked this problem "solved by humans" (11 August), because the authors said the human and AI contributions could not be cleanly separated, and relabeled it "human + AI" after changing its policy on 16 September. Until the result is reviewed and the AI's part is described by the authors themselves, we grade it "unverified". Problem description paraphrased from Epoch AI (CC BY 4.0).

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