Solution update

This problem has been solved by Huang et al. We hold a high bar for considering a problem “solved by AI”, in particular requiring that the core ideas of the solution be unambiguously contributed by AI. To judge this, we rely heavily on author attribution. In this case, one author is quoted as saying, “The boundaries between the human- and AI-contributed reasoning are not clear. Probably, it would be practically impossible to draw a sharp line.” Another author’s reflections state that, “Some of the most important decisions were clearly mathematical judgments made by the human collaborators.” We read this as saying it is not unambiguously clear that AI contributed the core ideas, and thus we do not consider this problem to be “solved by AI”.

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.

Prompt

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.