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