论文标题
编织轨道和Mathieu组$ M_ {23} $作为Galois Group
Braid orbits and the Mathieu group $M_{23}$ as Galois group
论文作者
论文摘要
目前，在$ \ mathbb {q} $上超过$ \ mathbb {Q} $的逆问题尚未解决Mathieu组$ M_ {23} $。在这里，使用刚度方法和编织的动作来实现$ m_ {23} $作为Galois组的当前状态的概述。计算$ M_ {23} $的计算辫子轨道除了Fried的举重不变性外，还揭示了辫子动作的新不变。这些不变式可用于构建通用的辫子轨道，以及在$ \ mathbb {q} $上的更多galois实现，用于Mathieu group $ m_ {24} $，但直到现在，直到现在，它并没有成功实现$ m_ {23} $，因为$ \ tho Mathbb {q} $ agalois galois galois group avalois Group。因此，$ m_ {23}/\ mathbb {q} $保持打开状态。最后，搜索合适的类向量的启发式方法，以实现小组的实现。
At present, the inverse Galois problem over $\mathbb{Q}$ is unsolved for the Mathieu group $M_{23}$. Here an overview of the current state in realizing $M_{23}$ as Galois group using the rigidity method and the action of braids is given. Computing braid orbits for $M_{23}$ revealed new invariants of the action of braids in addition to Fried's lifting invariant. These invariants can be used to construct generic braid orbits and more Galois realizations over $\mathbb{Q}$ for the Mathieu group $M_{24}$, but until now did not lead to success for realising $M_{23}$ as Galois group over $\mathbb{Q}$. Thus $M_{23}/\mathbb{Q}$ remains open. Finally, heuristics for searching suitable class vectors with regard to the realization of groups as Galois groups are given.
