Unramified Solutions of Inverse Galois Problems and their Applications to the Class Field Tower Problems

伽罗瓦反问题的无分支解及其在类场塔问题中的应用

• 批准号: 16540017
• 负责人: NOMURA Akito
• 金额: $ 1.6万
• 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: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540017/
• 关键词: embedding problem; unramified extension; class number; inverse Galois problem; イデアル類群; 類体論

Head investigator Nomura studied the existence of unramified 3-extensions over cyclic cubic fields. In particular we treated the following problem.Problem P(F,G) : For a given Galois extension F/Q and a finite group G, does there exists an unramified Galois extension M/F such that the Galois group Gal(M/F) is isomorphic to G.Let F and K be the cyclic cubic fields satisfying the certain ramification conditions. Let G_1 and G_2 be the non-abelian 3-group such that the GAP-number is [81,9] and [243,2] respectively. One of main results is stated as follows.Assume that the class number of F is divisible by 81. Then, (1) there exists an unramified extension M/K such that the Galois group Gal(M/K) is isomorphic to G_1, (2) there exists an unramified extension M/F such that the Galois group Gal(M/F) is isomorphic to G_2.We also studied the class number relation between certain cubic fields, and gave an alternative proof of the Naito's result.Investigator Hirabayashi constructed some multiple Dedekind sums and gave a relative class number formula for an imaginary abelian number field by means of such Dedekind sums. He also gave a generalization of Girstmair's formula to an imaginary abelian number field.

首席研究员野村（Nomura）研究了循环立方域上非分枝3-延拓的存在性。我们特别处理了以下问题。问题P(F，G)：对于给定的伽罗瓦扩展F/Q和有限群G，是否存在使伽罗瓦群Gal（M/F）同构于G的非分形伽罗瓦扩展M/F，设F和K为满足一定分形条件的循环三次场。设G_1和G_2为非阿贝尔3群，使得gap数分别为[81,9]和[243,2]。主要结果之一如下。假设F的类数能被81整除。则，(1)存在一个使伽罗瓦群Gal（M/K）与G_1同构的非分支扩展M/K，(2)存在一个使伽罗瓦群Gal（M/F）与G_2同构的非分支扩展M/F。我们还研究了某些三次域之间的类数关系，并给出了内藤结果的另一种证明。研究者Hirabayashi构造了一些多重Dedekind和，并利用这些Dedekind和给出了一个虚阿贝尔数域的相对类数公式。他还将格斯特梅尔公式推广到一个虚阿贝尔数域。

项目成果

Designs in a coset geometry Delsarte theory revisited

重温陪集几何 Delsarte 理论中的设计

• 发表时间: 2004
• 期刊: European J.Combin 25
• 作者: Masaaki Harada;Masaaki Kitazume and Akihiro Munemasa;内山 成憲;M.Oka;Akihiro Munemasa and Vladimir D.Tonchev;M.Oka;T.Ito
• 通讯作者: T.Ito

The shape of a tridiagonal pair

三对角对的形状

• 发表时间: 2004
• 期刊: J.Pure Appl 188
• 作者: Oka;M.;Hiroaki Terao;T.Ito and P.Algebra
• 通讯作者: T.Ito and P.Algebra

On Distributions of Multiple Access Interference for Spread Spectrum Communiction Systems Using M-Phase Spreading Sequences of Markov Chains

马尔可夫链M相扩频序列扩频通信系统多址干扰分布研究

• 发表时间: 2004
• 期刊: Proc.the 2004 IEEE Int.Symp.on Circuit and Systems
• 作者: Shunsuke Takagi;Kei-ichi Watanabe;Izumi MiyamotoA;H.Fujisaki
• 通讯作者: H.Fujisaki

中心拡大の埋め込み問題について

关于中心扩展嵌入问题

• 发表时间: 2004
• 期刊: 第2回北陸数論研究集会報告集
• 作者: Terasoma;T.;M.Hirabayashi;H.Fujisaki;H.Fujisaki;M.Hirabayashi;H.Fujisaki;A.Nomura
• 通讯作者: A.Nomura

On Bit Error Probabilities of SSMA Communication Systems Using Spreading Sequences of Markov Chains

马尔可夫链扩频序列SSMA通信系统误码率研究

• 发表时间: 2005
• 期刊: IEICE Transactions on Fundamentals E88・A
• 作者: Terasoma;T.;M.Hirabayashi;H.Fujisaki
• 通讯作者: H.Fujisaki