Professor Kyu-Hwan Lee, together with Xiaoyu Huang, Blake Jackson, Bjorn Poonen, Rachel Pries, and Shaowu Zhang, has resolved a long-standing case of the inverse Galois problem by proving that the Mathieu group $M_{23}$ occurs as a Galois group over the rational numbers. Their paper,“The Mathieu Group $M_{23}$ Is a Galois Group over $\mathbb{Q}$“, establishes the final missing sporadic simple group as a Galois group over $\mathbb{Q}$.The result is especially significant because $M_{23}$ was the last of the 26 sporadic finite simple groups not previously known to arise as a Galois group over the rational numbers. During the 1980s, mathematicians succeeded in realizing the other 25 sporadic groups, but the $M_{23}$ case resisted repeated attempts for decades. Lee and his collaborators have now completed this program.

The authors construct an explicit degree-23 polynomial with rational coefficients whose splitting field has Galois group $M_{23}$. To achieve this, they combine techniques from algebraic geometry, number theory, and computational mathematics, constructing an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$.

This breakthrough resolves an outstanding problem in inverse Galois theory.

Paper: The Mathieu Group $M_{23}$ Is a Galois Group over $\mathbb{Q}$ [arxiv.org]