The Mathieu Group M23 as a Galois Group over Q
The case of the inverse Galois problem is resolved: the Mathieu group occurs as a Galois group over . More strongly, there exists a regular Galois extension of with Galois group , and hence infinitely many -extensions of . This completes the realization over of all sporadic finite simple groups. The general inverse Galois problem remains open.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Inverse Galois theory
- Posed by
- —
- Year posed
- —
- Years open
- —
- Solved
- 2026-08-08
- Model
- Claude Fable 5; Claude Opus 4.8; ChatGPT 5.6 Sol
- Vendor
- Anthropic; OpenAI
- Collaborators
- Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, Shaowu Zhang
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 55 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Resolves the M23 case, not the general inverse Galois problem.
What the AI did
The authors used Claude Fable 5, Claude Opus 4.8, and ChatGPT 5.6 Sol for literature search, code generation, testing hypotheses, ruling out unsuccessful approaches, devising computational strategies, and checking results. The project began at the May 27–30, 2026 AIM workshop “AI and number theory.” The authors state that no article text was written by AI and that the final results were verified independently of AI using Magma and PARI/GP.
Verification
August 2026 preprint; no independent expert review or peer review identified as of submission. The authors report that the final results were verified without AI using Magma and PARI/GP. In particular, the explicit degree-23 polynomial is computationally certified to have splitting field with Galois group M23. Numerical Belyi-map computations are used in constructing the cover, with algebraic recognition and subsequent exact verification. The authors provide their computational code publicly.
Sources
Submitted by HiddenHawk615 on