A computational research on number fields and coverings with non-commutative Galois groups

2014 Fiscal Year Final Research Report

• Project/Area Number: 22540032
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Sophia University
• Principal Investigator: TSUNOGAI Hiroshi 上智大学, 理工学部, 教授 (20267412)
• Co-Investigator(Kenkyū-buntansha): TSUZUKI Masao 上智大学, 理工学部, 准教授 (80296946) ; GOMI Yasushi 上智大学, 理工学部, 准教授 (50276515) ; UMEGAKI Atsuki 愛知大学, 国際コミュニケーション学部, 准教授 (60329109) ; KOMATSU Toru 東京理科大学, 理工学部, 講師 (10403974) ; NAKASUJI Maki 上智大学, 理工学部, 准教授 (30609871)
• Co-Investigator(Renkei-kenkyūsha): RIKUNA Yuichi 電気通信大学, 情報理工学部, 准教授 (10434309) ; HOSHI Akinari 立教大学, 理学部, 助教 (50434262)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: 整数論 / 代数学 / ガロア理論 / アルゴリズム

Outline of Final Research Achievements

Our research intends to study Galois extensions with non-commutative Galois groups concretely. On Cross-Ratio Noether Problem for transitive groups of degree six, we obtained affirmative answers with explicit generators except two non-solvable subgroups. On Relative Rationality between the fixed fields of the fields of cross-ratios and the fields of ratios of differences, for the cases of the symmetric groups, it was shown to be affirmative for odd degrees and to be negative for even degrees. We also obtained negative answers for some subgroups of the symmetric groups of even degrees. We determined the defining equations of the dessins d'enfants of genus one of degree six except one valency list, and confirmed that known Galois invariants of dessins distinguish their Galois orbits in these cases. We showed the existence and non-existence of number fields with given prime inert conditions by means of generic polynomials.

Free Research Field

整数論