AI Solved Final Unsolved Sporadic Galois Group

Researchers utilized artificial intelligence to define an explicit polynomial for the sporadic group M23.

Updated on Sept. 22, 2026 in Mathematics

Bold flat-color editorial illustration showing an abstract, geometric manifold in navy and orange, representing a mathematical symmetry proof. — Researchers successfully solved the inverse Galois problem for the sporadic group M23, finalizing a decades-long pursuit in algebraic symmetry. AI Illustration. Upload story photo >

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In May 2026, researchers successfully solved the inverse Galois problem for the sporadic group M23, the final remaining case among 26 such groups. This achievement marks a transition from long-standing open problems in algebra to verifiable, computer-assisted results.

Why it matters

Solving this problem helps mathematicians map the absolute Galois group, a fundamental structure that characterizes the symmetries of all polynomials. This success demonstrates how AI-driven search methods can overcome computational barriers that have persisted since the 1980s.

The team utilized AI to approximate surfaces and construct an explicit degree-23 polynomial for the M23 group using 90 digits of precision. This effort was supported by a crowdsourced competition that identified 25,000 relationships between polynomials and groups acting on 24 roots.

The players

Rachel Pries

A researcher affiliated with Colorado State University known for expertise in arithmetic geometry and algebraic structures.

Bjorn Poonen

A researcher at the Massachusetts Institute of Technology focused on number theory and algebraic geometry.

Xiaoyu Huang

A mathematician at Temple University working on the intersection of algebra and computational methods.

Kyu-Hwan Lee

A researcher at the University of Connecticut specializing in Lie theory and algebraic combinatorics.

Jen Paulhus

An organizer at Mount Holyoke College who facilitates research in algebraic curves and symmetry groups.

The details

The inverse Galois problem asks whether every Galois group—a mathematical structure representing the symmetries of a polynomial's roots—corresponds to a specific polynomial. By using AI to search for combinations of symmetries and approximate the necessary algebraic surfaces, the six-person research team successfully constructed the required equation. Participants in the supporting competition further refined these techniques by combining human calculation with AI-driven approximation methods.

May 2026: Mathematicians gathered at the California Institute of Technology to discuss the research.
Late August 2026: The first phase of the competition concluded.

The Tech Race

This finding completes the inverse Galois problem for all 26 sporadic groups, closing a gap left by the 1980s classification efforts. It establishes a new benchmark for using AI to search for and verify high-degree polynomials that were previously computationally inaccessible.

This development represents a shift in methodology for pure mathematics, indicating that AI-assisted search will increasingly serve as a primary tool for solving complex algebraic problems. Researchers and graduate students in number theory will likely adopt these specific AI-driven approximation workflows to approach other open problems in symmetry.

The takeaway

The solution for M23 confirms that AI can effectively bridge the gap between abstract group theory and explicit polynomial construction. Mathematicians should track future developments in the absolute Galois group research program to see if these methods scale beyond sporadic cases.