New developments in number theory from the Kummer theory of algebraic tori

• Project/Area Number: 19540015
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: The University of Electro-Communications
• Principal Investigator: KIDA Masanari The University of Electro-Communications, 大学院・情報理工学研究科, 教授 (20272057)
• Co-Investigator(Kenkyū-buntansha): OHNO Masahiro 電気通信大学, 大学院・情報理工学研究科, 准教授 (70277820)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 数論 / クンマー理論 / 逆ガロア問題 / 代数的トーラス / イデアル類群 / 代数的整数論 / ガロア理論 / 代数群 / クンマー拡大 / 生成的多項式 / 巡回拡大 / Brumerの多項式

Research Abstract

In this research project, we are primarily concerned with a duality between the groups of rational points of algebraic tori and the Galois groups of maximal elementary abelian extensions over base fields without roots of unity. This is a natural generalization of the classical Kummer theory. We also compute generic polynomials for cyclic groups of small order in the case where the base field is a prime field. Several applications are also studied.

Research Products

• [Journal Article] A Kummer theoretic construction of an S_3-polynomial with given quadratic subfield. (2010)
  • Author(s): Kida, Masanari
  • Journal Title: Interdisciplinary Information Sciences 16 Pages: 17-20
• [Journal Article] Classifying Brumer's quintic polynomials by weak Mordell-Weil groups. (2010)
  • Author(s): Kida, Masanari, Rikuna, Yuichi, Sato, Atsushi
  • Journal Title: Int.J.Number Theory. 6 Pages: 691-704
• [Journal Article] Descent Kummer theory via Weil restriction of multiplicative groups. (2010)
  • Author(s): Kida, Masanari
  • Journal Title: J.Number Theory 130 Pages: 639-659
• [Journal Article] Classifying Brumer's quintic polynomials by weak Mordell Weil groups (2010)
  • Author(s): Masanari Kida, 他
  • Journal Title: International J.Number Theory Volume: 6 Pages: 691-704
  • Peer Reviewed
• [Journal Article] A Kummer-theoretic construction of an S_3 polynomial with given quadratic subfields (2010)
  • Author(s): Masanari Kida
  • Journal Title: Interdisciplinary Information Sciences Volume: 16 Pages: 17-20
  • Peer Reviewed
• [Journal Article] Descent Kummer theory via Weil restriction of multiplicative groups (2010)
  • Author(s): Kida, Masanari
  • Journal Title: J.Number Theory 130 Pages: 639-659
  • Peer Reviewed
• [Journal Article] Descent Kummer theory via Weil restriction of multiplicative groups (2010)
  • Author(s): Masanari Kida
  • Journal Title: Journal of Number Theory 130 Pages: 639-659
  • Peer Reviewed
• [Journal Article] Quintic polynomials of Hashimoto-Tsunogai, Brumer, and Kummer. (2009)
  • Author(s): Kida, Masanari, Renault, Guenael Yokoyama, Kazuhiro
  • Journal Title: Int.J.Number Theory 5 Pages: 555-571
• [Journal Article] Kazuhiro : Quintic polynomials of Hashimoto-Tsunogai, Brumer, and Kummer (2009)
  • Author(s): Kida, Masanari, Renault, Guenael, Yokoyama
  • Journal Title: Int. J. Number Theory 5 Pages: 555-571
  • Peer Reviewed
• [Journal Article] D5 拡大のクンマー理論 (2008)
  • Author(s): 木田雅成
  • Journal Title: 早稲田大学整数論研究集会報告集 13 Pages: 51-62
• [Presentation] 初心者のためのMagma入門 (2010)
  • Author(s): 木田雅成
  • Organizer: Magmaでひろがる数学の世界
  • Place of Presentation: 九州大学
  • Year and Date: 2010-10-09
• [Presentation] t構造つき三角圏でのtitltingについて (2010)
  • Author(s): 大野真裕
  • Organizer: 数理科学セミナー
  • Place of Presentation: 高知大学理学部
  • Year and Date: 2010-09-14
• [Presentation] イデアル類群の鏡映定理について (2010)
  • Author(s): 木田雅成
  • Organizer: 香川セミナー
  • Place of Presentation: 香川大学教育学部
  • Year and Date: 2010-06-12
• [Presentation] Metacyclic拡大の構成について (2009)
  • Author(s): 木田雅成
  • Organizer: 東北大学代数セミナー
  • Place of Presentation: 東北大学理学研究科
  • Year and Date: 2009-09-08
• [Presentation] Constructing metacyclic extensions. (2009)
  • Author(s): 木田雅成
  • Organizer: First PRIMA Congress
  • Place of Presentation: The University of New South Wales, Sydney, Australia.
  • Year and Date: 2009-07-10
• [Presentation] Constructing metacyclic extensions. First PRIMA Congress. (2009)
  • Author(s): Masanari Kida
  • Organizer: The University of New South Wales
  • Place of Presentation: Sydney, Australia
  • Year and Date: 2009-07-10
• [Presentation] Constructing metacyclic extensions (2009)
  • Author(s): Masanari Kida
  • Organizer: 1^<st> Prima Congress
  • Place of Presentation: University of New South Wales, Australia
  • Year and Date: 2009-07-10
• [Presentation] 数論研究者のためのMagma入門 (2008)
  • Author(s): 木田雅成
  • Organizer: 第7回北陸数論研究集会
  • Year and Date: 2008-12-26
• [Presentation] 計算代数システムMagmaによる代数構造の計算 (2008)
  • Author(s): 木田雅成
  • Organizer: 代数的整数論とその周辺
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-12-10
• [Presentation] On metacyclic extensions. (2008)
  • Author(s): 木田雅成
  • Organizer: 日韓整数論セミナー
  • Place of Presentation: 東北大学片平キャンパスさくらホール
  • Year and Date: 2008-11-14
• [Presentation] On metacyclic extensions. (2008)
  • Author(s): 木田雅成
  • Organizer: 研究集会「ガロア理論とその周辺徳島2008」
  • Place of Presentation: 徳島大学工業会館
  • Year and Date: 2008-09-09
• [Presentation] On metacyclic extensions (2008)
  • Author(s): 木田雅成
  • Organizer: 研究集会「ガロア理論とその周辺徳島2008」
  • Place of Presentation: 徳島大学工業会館
  • Year and Date: 2008-09-09
• [Presentation] Classification of Brumer's quintic dihedral polynomials. (2008)
  • Author(s): 木田雅成
  • Organizer: Foundation of Computational Mathematics (FoCM08) Computational Number Theory Workshop City
  • Place of Presentation: University of Hong Kong
  • Year and Date: 2008-07-25
• [Presentation] Classification of Brumer's quintic dihedral polynomials (2008)
  • Author(s): Masanari Kida
  • Organizer: Foundation of Computational Mathematics (FoCM08) Computational Number Theory Workshop
  • Place of Presentation: City University of Hong Kong
  • Year and Date: 2008-06-25
• [Presentation] Classification of Brumer's quintic polynomials (2008)
  • Author(s): Masanari Kida
  • Organizer: Foundation of Computational Mathematics
  • Place of Presentation: City University of Hong Kong
  • Year and Date: 2008-06-25
• [Presentation] D_5拡大のクンマー理論 (2008)
  • Author(s): 木田雅成
  • Organizer: 第13回早稲田大学整数論研究集会
  • Place of Presentation: 早稲田大学理工学部
  • Year and Date: 2008-03-11
• [Presentation] A classification of isomorphic classes of Brumer's quintic polynomials using elliptic curves. (2007)
  • Author(s): 木田雅成
  • Organizer: 研究集会「ガロア理論とその周辺」
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2007-11-03
• [Presentation] A classification of isomorphic classes of Brumer's quintic polynomials using elliptic curves (2007)
  • Author(s): 木田雅成, 陸名雄一
  • Organizer: 研究集会「ガロア理論とその周辺」
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2007-11-03
• [Presentation] Brumerの多項式の分解体について (2007)
  • Author(s): 木田雅成、横山和弘, Guenael Renault
  • Organizer: 日本数学会2007年度秋季総合分科会
  • Place of Presentation: 東北大学理学部
  • Year and Date: 2007-09-23
• [Presentation] Brumer の多項式の分解体について (2007)
  • Author(s): 木田雅成
  • Organizer: 日本数学会2007年度秋季総合分科会
  • Place of Presentation: 東北大学理学部
  • Year and Date: 2007-09-23
• [Presentation] Descent Kummer theory via Weil restriction of multiplicative groups. (2007)
  • Author(s): 木田雅成
  • Organizer: 25th Journees Arithmetiques
  • Place of Presentation: University of Edinburgh, Scotland.
  • Year and Date: 2007-07-03
• [Presentation] Descent Kummer theory via Weil restriction of multiplicative groups (2007)
  • Author(s): 木田 雅成
  • Organizer: 25th Journees Arithmetiques
  • Place of Presentation: University of Edinburgh
  • Year and Date: 2007-07-03
• [Presentation] Kummer theory without roots of unity. (2007)
  • Author(s): 木田雅成
  • Organizer: Seminaire de calcul formel SPIRAl/SALSA
  • Place of Presentation: Universite Pierre et Marie Curie, Paris, France.
  • Year and Date: 2007-06-27
• [Presentation] 乗法群のヴェイユ制限による降下クンマー理論 (2007)
  • Author(s): 木田雅成
  • Organizer: 早稲田整数論セミナー
  • Year and Date: 2007-03-18
• [Remarks] ホームページ等
  • URL: http://mathweb.e-one.uec.ac.jp/~kida/index.html
• [Remarks]
  • URL: http://mathweb.e-one.uec.ac.jp/~kida
• [Remarks]
  • URL: http://mathweb.e-one.uec.ac.jp/~kida/math/mypaper.html