Mathematicians Solved Inverse Galois Problem for M23
Researchers utilized artificial intelligence to construct a degree-23 polynomial for the final sporadic group.
Updated on Sept. 22, 2026 in Mathematics
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In May 2026, a research team successfully resolved the inverse Galois problem for the sporadic group M23. This achievement marked the final piece of a puzzle involving 26 sporadic groups that had been studied since the 1980s.
Why it matters
Solving this problem helps mathematicians map the absolute Galois group, which acts as a fundamental descriptor for the symmetries found in all polynomials. This accomplishment brings closure to a long-standing challenge in algebraic group theory.
The team constructed an explicit degree-23 polynomial for M23 using 90 digits of numerical precision. This process involved approximating a set of 7 surfaces linked to the group.
The players
Rachel Pries
She is a researcher affiliated with Colorado State University who contributed to the study.
Bjorn Poonen
He is a mathematician at the Massachusetts Institute of Technology involved in the inverse Galois research.
Xiaoyu Huang
He is a researcher based at Temple University who participated in the project.
Kyu-Hwan Lee
He is a mathematician at the University of Connecticut who collaborated on the team.
Jen Paulhus
She is a professor at Mount Holyoke College who organized the research competition.
The details
Researchers utilized AI tools to search for complex combinations of symmetries and approximate the necessary surface equations. A parallel crowdsourced competition hosted by the Foundation for Science and AI Research further supported this effort by documenting 25,000 unique relationships between polynomials and groups acting on 24 roots.
Timeline
May 2026: Mathematicians gathered at Caltech for the initial research sprint.
Late August 2026: The first phase of the global crowdsourced competition concluded.
The Big Picture
This discovery concludes the task set by the classification of 25 sporadic groups completed by the 1980s. By finally solving M23, the researchers effectively bridge a significant gap in the structural understanding of the absolute Galois group.
This mathematical breakthrough advances the computational methods available for studying polynomial symmetries, which could eventually refine algorithms used in cryptography and data security. While theoretical, the integration of AI into these proofs suggests a shift toward faster verification of complex algebraic structures.
The takeaway
The successful application of AI to solve the M23 inverse Galois problem illustrates how human-machine collaboration can resolve long-standing theoretical bottlenecks. Researchers can apply these new computational strategies to simplify the mapping of complex symmetry groups in future studies.
Further reading
For more on the developments in algebraic group theory, visit the Mathematics section.
Source note: This article includes information reported by Scientific American.
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