Inverse Galois

Find a polynomial whose Galois group is the Mathieu group $M_{23}$.

Notability: Major advance

Solved: No

Contributor: Epoch Staff

Field: Number Theory

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About the problem Attempts by AI AI prompts Mathematician survey

About the problem

The inverse Galois problem asks whether every finite group is the Galois group of some extension of the rational numbers. In any given case, this amounts to finding a polynomial with prescribed symmetries. The aim of this problem is to find a polynomial whose Galois group is the Mathieu group $M_{23}$. This is an especially interesting case because it is the last of the sporadic simple groups for which no such polynomial is known.

It is not guaranteed that such a polynomial must exist, but mathematicians generally expect it to.

Warm-up: we ask for a polynomial whose Galois group is the Mathieu group $M_{22}$. Such constructions are well-known.

Attempts by AI

We have evaluated the following models on this problem. “Warm-up” refers to an easier variant of the problem with a known solution.

Model

Problem

Solved

Logs

GPT-5.2 Pro

Warm-up

Yes

GPT-5.2 Pro

Full Problem

Gemini 3 Deep Think

Full Problem

AI Prompts

Warm-up

Find a degree 22 polynomial in Z[x] whose splitting field over Q has Galois group M_{22}. The coefficients of the polynomial must have fewer than 100 decimal digits.

Provide your solution as a string in Magma syntax, e.g. 3*x^2 - 2*x + 1.

Full problem

Find a degree 23 polynomial in Z[x] whose splitting field over Q has Galois group M_{23}. The coefficients of the polynomial must have fewer than 100 decimal digits.

Provide your solution as a string in Magma syntax, e.g. 3*x^2 - 2*x + 1.

Mathematician survey

The author assessed the problem as follows.

# of mathematicians highly familiar with the problem: a majority of those working in a subfield (≈100)

# of mathematicians who have made a serious attempt to solve the problem: 5–10

Rough guess of how long it would take an expert human to solve the problem: 3–12 months

Notability of a solution: a major advance in a subfield

A solution would be published: in a strong specialty journal

Likelihood of a solution generating more interesting math: fairly likely: the problem is rich enough that most solutions should open new avenues

Probability that the problem is solvable as stated: 95-99%