Arithmetic of constructive inverse Galois problem

• Project/Area Number: 24540011
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo University of Science (2013-2014); The University of Electro-Communications (2012)
• Principal Investigator: KIDA Masanari 東京理科大学, 理学部, 教授 (20272057)
• Co-Investigator(Kenkyū-buntansha): OHNO Masahiro 電気通信大学, 情報理工学研究科, 准教授 (70277820)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: ガロア理論 / 生成多項式 / 代数的トーラス

Outline of Final Research Achievements

In this research we aimed at constructing families of polynomials with given finite groups as Glois group.
In particular, we were interested in families with arithmetic information such as ramification and ideal decomposition law. Our first object was Kummer extensions associated to algebraic tori. We made extensive computation on low dimensional tori case to find arithmetic information. I believe that it will soon become possible to get a generalization for certain tori of arbitrary dimensions. We obtained a new family of cyclic polynomials by using algebraic tori of norm tori. I submitted an article concerning this construction. I also made some investigation concerning dihedral extensions in the relation of complex multiplication.

Research Products

• [Journal Article] Involutions on generating functions (2013)
  • Author(s): Masanari Kida and Yuichiro Urata
  • Journal Title: Journal of Integer Sequences Volume: 16
  • Peer Reviewed
• [Presentation] 相対ノルムの代数的トーラスの同種写像について (2015)
  • Author(s): 木田雅成
  • Organizer: 愛知数論セミナー
  • Place of Presentation: 愛知工業大学 本山キャンパス
  • Year and Date: 2015-01-31
• [Presentation] Involutions on generating functions (2013)
  • Author(s): Masanari Kida
  • Organizer: Journees Arithmetiques
  • Place of Presentation: Institut Fourier, Grenoble, France
• [Presentation] 巡回多項式族の構成とその数論 (2012)
  • Author(s): 木田雅成
  • Organizer: 埼玉大学談話会
  • Place of Presentation: 埼玉大学
  • Invited