AI Solves Decades‑Old Inverse Galois Problem

A team of mathematicians, guided by AI, cracked the inverse Galois problem for the M23 sporadic group, revealing a degree‑23 polynomial whose symmetries match the group. This breakthrough advances the quest for the absolute Galois group and demonstrates AI’s power in pure math.

By Felo News Desk · Published Tue, 22 Sep 2026 12:17:42 GMT

In May, a gathering of mathematicians at the California Institute of Technology’s tallest building sparked a collaboration that would resolve a problem that had stumped scholars for more than a century. The event, organized by the American Institute of Mathematics, invited participants to propose questions that could benefit from artificial intelligence’s ability to scan vast mathematical landscapes. Rachel Pries of Colorado State University presented one such question: the inverse Galois problem for the M23 sporadic group.

What the Inverse Galois Problem Is

Algebraic equations called polynomials can have roots that are rational, irrational, or even imaginary. When the roots of a polynomial are shuffled in all possible ways that still satisfy the equation, the set of these shuffles forms a mathematical object known as a Galois group. The classic work of Évariste Galois in the 19th century classified many such groups, but a lingering question remained: given a particular Galois group, can we always find a polynomial whose roots exhibit exactly that symmetry?

This question is the inverse Galois problem. While algorithms exist to determine the Galois group of a known polynomial, the reverse—constructing a polynomial from a desired group—is notoriously difficult. Most groups fit into orderly families, but 26 so‑called sporadic groups, including M23, resist such classification.

From a Conference Idea to a Dream Team

After the Caltech talks, participants were asked to bid on problems to tackle. The inverse Galois problem for M23 attracted the most interest, and six mathematicians formed a new team: Pries, Bjorn Poonen (MIT), Xiaoyu Huang (Temple University), Blake Jackson (Institute for Computer‑Aided Reasoning in Mathematics), Kyu‑Hwan Lee (University of Connecticut), and Shaowu Zhang (Caltech). Within three months, they announced that they had found a polynomial whose Galois group is M23.

Their approach combined human insight with AI. They first used AI to search for combinations of symmetries within M23, identifying a minimal set of seven geometric surfaces. The AI then produced high‑precision numerical approximations of these surfaces. When the team reached 90 decimal digits, computational limits forced a pause. Huang suggested that the choice of coordinates might be the obstacle, prompting the team to deploy a swarm of AI agents to explore alternative coordinate systems.

The AI‑Assisted Breakthrough

One AI agent returned with a promising new coordinate configuration. The team refined the approximation, eventually deriving an explicit degree‑23 polynomial whose roots shuffle exactly as M23 dictates. The polynomial, displayed in full in the original article, is a monumental piece of algebraic data that confirms the existence of a polynomial for the M23 group.

This result is more than a single polynomial; it is a step toward understanding the “absolute Galois group,” a theoretical construct that would describe the symmetries of all polynomials simultaneously. Each new polynomial found for a sporadic group brings mathematicians closer to mapping this grand structure.

Parallel Efforts and the Role of AI

While the team focused on M23, a separate crowdsourced competition launched by the Foundation for Science and AI Research (SAIR) invited participants worldwide to find polynomials for all 24‑root groups, including the symmetric group S24. The competition attracted both professional number theorists and hobbyists, many of whom used AI to aid their searches. Although AI helped streamline submissions, the winning team—two German mathematicians—credited human ingenuity as the decisive factor.

These parallel efforts illustrate how AI can accelerate discovery in pure mathematics, yet human creativity remains essential for interpreting results and guiding the search.

Looking ahead, the newly discovered polynomial for M23 will serve as a benchmark for testing computational techniques and for exploring deeper connections within Galois theory. The broader quest for the absolute Galois group continues, with each sporadic group’s polynomial offering a new piece of the puzzle.

Why This Matters

Solving the inverse Galois problem for M23 not only resolves a long‑standing question but also demonstrates the practical power of AI in advancing theoretical mathematics. This breakthrough could pave the way for new methods in cryptography, coding theory, and other fields that rely on polynomial symmetries.

Key takeaways from this story include the successful collaboration between human mathematicians and AI, the importance of sporadic groups in understanding the absolute Galois group, and the growing role of crowdsourced competitions in mathematical research.

As AI tools become more sophisticated, their integration into mathematical research promises to unlock further mysteries of the abstract world, bringing us closer to a unified understanding of polynomial symmetries.

For those interested in the technical details, the full polynomial is available in the original publication and can be examined for its intricate structure of 23 roots and the M23 symmetry it embodies.

Key facts

AI guided mathematicians to solve the inverse Galois problem for the elusive M23 group.
A team of six researchers combined human insight with AI to find a degree‑23 polynomial.
The result advances the quest for the absolute Galois group, a central goal in number theory.
Parallel crowdsourced competitions demonstrate AI’s supportive role in mathematical discovery.
The breakthrough highlights the importance of sporadic groups in understanding finite simple groups.

Why it matters

The discovery confirms a decades‑old conjecture, showcasing how AI can accelerate pure mathematical research and bringing us closer to a comprehensive understanding of polynomial symmetries.

Frequently asked questions

What is the inverse Galois problem?

It asks whether, for any given Galois group, there exists a polynomial whose roots exhibit exactly that group’s symmetries.

Why is M23 special?

M23 is one of 26 sporadic groups that do not fit into the standard families of finite simple groups, making it a particularly challenging target.

How did AI help?

AI scanned vast symmetry combinations, generated high‑precision numerical approximations, and suggested new coordinate systems that led to the explicit polynomial.

What is the absolute Galois group?

It is a theoretical construct that would encapsulate the symmetries of all polynomials, representing the ultimate goal of Galois theory.

Will this affect everyday technology?

While the result is fundamental, insights from Galois theory can influence cryptography and error‑correcting codes, which underpin secure communications.