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.
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%