An Algorithm of Katz and its Application to the Inverse Galois Problem
An Algorithm of Katz and its Application to the Inverse Galois Problem
复制标题
Katz算法及其在伽罗瓦反问题中的应用
DOI：
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发表时间：
2000
影响因子：
0.7
通讯作者：
Stefan Reiter
中科院分区：
文献类型：
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作者：
M. Dettweiler;Stefan Reiter
在本文中，我们提出了一个新的和基本的方法来证明Katz（1996）的主要结果使用约旦？Takano and Bannai（1976）和Haraoka（1994）的Pochhammer矩阵。我们找到了Katz（1996）的中卷积的一个显式版本，它连接线性群中的某些矩阵元组。从这里，卡茨？线性群中刚性元组的存在性算法可以很容易地推导出来。可以进一步证明，元组上的卷积运算与辫群作用是可交换的。这在逆伽罗瓦理论中产生了一种新的方法，用于将线性群的子群正则地实现为Q上的伽罗瓦群。然后应用这种方法将许多系列的经典群规则地实现为Q上的伽罗瓦群。在附录中，我们处理卷积的加法版本。
In this paper we present a new and elementary approach for proving the main results of Katz (1996) using the Jordan?Pochhammer matrices of Takano and Bannai (1976) and Haraoka (1994). We find an explicit version of the middle convolution of Katz (1996) that connects certain tuples of matrices in linear groups. From this, Katz? existence algorithm for rigid tuples in linear groups can easily be deduced. It can further be shown that the convolution operation on tuples commutes with the braid group action. This yields a new approach in inverse Galois theory for realizing subgroups of linear groups regularly as Galois groups over Q. This approach is then applied to realize numerous series of classical groups regularly as Galois groups over Q. In the Appendix we treat an additive version of the convolution.