Solution update

First solved with: Claude Fable 5, Claude Opus 4.8, ChatGPT 5.6 Sol

This problem has been solved by Huang et al. While we initially marked this problem as being solved by humans, we have since introduced a new “human + AI” solution status. Since comments from the authors make it clear that AI was instrumental in solving this problem, we have updated its status with this new label.

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 M23. The coefficients of the polynomial must have fewer than 100 decimal digits.

Provide your solution as a single polynomial expression in Magma syntax, e.g.
  3*x^2 - 2*x + 1
Only integer literals, the variable x (lower case), the operators + - * ^ and parentheses are
accepted. Do not wrap the expression in quotation marks, and do not add a trailing semicolon, an
assignment, or any other Magma code.