Mathematicians use AI to solve decades-old inverse Galois problem

September 22, 2026 at 04:00 PM1 min

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Mathematicians at a California Institute of Technology workshop used AI to solve the inverse Galois problem, a decades-old challenge linking polynomial equations to symmetry groups. The breakthrough came within months of the May meeting, where Colorado State University mathematician Rachel Pries presented the problem. The solution intertwines human insight with AI's ability to explore vast mathematical possibilities.

Key Facts

The inverse Galois problem asks whether every finite group of symmetries arises as the Galois group of some polynomial equation with rational coefficients.
Rachel Pries, a mathematician at Colorado State University, presented the problem at a May workshop at the California Institute of Technology.
The problem is inspired by the work of 19th-century French mathematician Évariste Galois, who died at age 20 after a gun duel.
Most Galois groups fall into orderly families, but 26 exceptional groups do not fit the standard ladder.

The Inverse Galois Problem

The inverse Galois problem asks whether every finite group of symmetries can be realized as the Galois group of some polynomial equation with rational coefficients. For any given polynomial, known algorithms can determine its Galois group, but the reverse direction—finding a polynomial for a given group—remained opaque. The problem traces back to Évariste Galois, who classified polynomials by the symmetries of their solutions before dying in a duel at age 20. Most Galois groups form orderly families indexed by the number of roots, but 26 exceptional groups refuse to fit this pattern.

AI-Assisted Breakthrough

At a May workshop at the California Institute of Technology, Rachel Pries of Colorado State University presented the inverse Galois problem as a candidate for AI-assisted exploration. The American Institute of Mathematics solicited problems suited to AI's ability to search through vast combinatorial possibilities. Within a few months, mathematicians combined human insight with AI scale to solve the decades-old problem.