Inverse Galois Problems with Restricted Ramifications

有限分支的逆伽​​罗瓦问题

• 批准号: 14540018
• 负责人: NOMURA Akito
• 金额: $ 1.73万
• 依托单位: KANAZAWA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540018/
• 关键词: embedding problem; unramified extension; class number; class field tower; inverse Galois problem

Our research results are summarized as follows. Head investigator Nomura studied the unramified solution of embedding problems and the existence of unramified p-extensions. One of main results in case when p is an odd prime is stated as follows. Let G be the group such that the GAP-number is[243,65], which is a non-abelian 3-group of order 243. If the class number of quintic cyclic fields F is divisible by 3,then there exists an unramified Galois extension over F such that the Galois group is isomorphic to G. In particular, the class number of the Hilbert class field of F is divisible by 3. In case when p=2,we also studied the existence of unramified quaternion extension over cyclic fields, and gave an affirmative answer of a special case of Fontaine-Mazur-Boston conjecture concerning the Galois group of class field tower.Investigator Morishita discussed some analogies for primes coming from link theory, based on an analogy between the structure of the group of a link and those of certain Galois group. He also gave a cohomological interpretation of Redei's symbol by using refined Milnor invariants, and generalized a classical results of Redei concerning the ideal class group of quadratic fields.

我们的研究结果总结如下。首席研究员野村研究了嵌入问题的非分枝解和非分枝p-扩展的存在性。当p是奇素数时的一个主要结果如下。设G为使gap -数为[243,65]的群，是一个243阶的非阿贝尔3-群。如果五次循环域F的类数可被3整除，则在F上存在一个非分支伽罗瓦扩展，使得伽罗瓦群与g同构，特别是F的Hilbert类域的类数可被3整除。当p=2时，我们还研究了循环域上非分支四元数可拓的存在性，并给出了关于类域塔伽罗瓦群的Fontaine-Mazur-Boston猜想的一个特例的肯定回答。研究者Morishita在链接群结构与伽罗瓦群结构的类比基础上，讨论了来自链接理论的素数的类比。他还利用改进的Milnor不变量对Redei符号进行了上同调解释，并推广了关于二次域理想类群的Redei经典结果。

项目成果

A.Nomura: "A note on unramified quaternion extensions over quadratic number fields"Proc.Japan Acad.. 78. 80-82 (2002)

A.Nomura：“关于二次数域上的未分支四元数扩展的注释”Proc.Japan Acad.. 78. 80-82 (2002)

T.Ito: "The shape of a tridiagonal pair"Journal of Pure and Applied Algebra. 印刷中.

T.Ito：“三对角对的形状”《纯粹与应用代数杂志》正在出版。

T.Ito: "The shape of a tridiagonal pair"Journal of Pure and Applied Algebra. (to appear).

T.Ito：“三对角对的形状”纯粹与应用代数杂志。

T.Ito: "The shape of a tridiagonal pair"Journal of Pure and Applied Algebra. (印刷中). (2004)

T.Ito：“三对角对的形状”《纯粹与应用代数杂志》（出版中）。

A.Nomura: "Notes on the existence of certain unramified 2-extensions"Illinois Journal of Math.. (印刷中).

A. Nomura：“关于某些无分支 2-扩展的存在的注释”伊利诺伊州数学杂志（正在出版）。