发表机构
Temple University; Carnegie Mellon University; University of Connecticut; Korea Institute for Advanced Study; Massachusetts Institute of Technology; Colorado State University; California Institute of Technology（天普大学; 卡内基梅隆大学; 康涅狄格大学; 韩国高等研究院; 麻省理工学院; 科罗拉多州立大学; 加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究完成逆伽罗瓦问题的相关计划，证明Mathieu群$M_{23}$是$\boldsymbol{\text{Q}}$上的伽罗瓦群，构造了对应显式23次多项式，利用非刚性共轭类三元组与数值Belyi映射算法完成证明。
AI 中文摘要
研究逆伽罗瓦问题的学者在1984至1989年间，已将26个零散有限单群中的25个实现为$\boldsymbol{\text{Q}}$上的伽罗瓦群。我们完成该计划，证明最后一个剩余的零散群Mathieu群$M_{23}$是$\boldsymbol{\text{Q}}$上的伽罗瓦群。实际上，我们构造了一个具有有理系数的显式23次多项式，其分裂域在$\boldsymbol{\text{Q}}$上的伽罗瓦群为$M_{23}$。为完成此工作，我们利用$M_{23}$的一个非刚性共轭类三元组，计算Belyi映射以构造$\boldsymbol{\text{Q}}(t)$上具有伽罗瓦群$M_{23}$的显式正则伽罗瓦扩张。我们计算的关键是Klug、Musty、Schiavone、Sijsling和Voight开发并实现的数值Belyi映射算法，该算法灵感来源于Hejhal和Stark的思想。
英文摘要
Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984-1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\mathbb{Q}$. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of $M_{23}$. We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Costa, Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an $M_{22}$-extension of $\mathbb{Q}$.
CommentsLots of additional information, 22 pages. Simpler equation for the regular M_23-extension of Q(t). More info about its specializations (bonus: a new realization of M_22 as a Galois group over Q). More info about coefficient recognition and rigorous Galois group computation. More info about the other six regular M_23-extensions over a degree 6 field arising from this Nielsen class
