Study of real quadratic fields by using continued fractions

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04779
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Aichi University of Education
• Principal Investigator: Kishi Yasuhiro 愛知教育大学, 教育学部, 准教授 (60380375)
• Co-Investigator(Kenkyū-buntansha): 冨田 耕史 名城大学, 理工学部, 准教授 (50300207)
• Research Collaborator: KAWAMOTO Fuminori
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 数論 / 連分数 / 二次体 / イデアル類群 / 類数

Outline of Final Research Achievements

The present research has mainly dealt with the continued fraction expansions of certain quadratic irrationals. The main results have been to find some properties of that of the minimal elements with even period and to obtain some relations between them. Moreover, we gave a lower bound for the class numbers of certain real quadratic fields by using the class number formula and the Yokoi invariant. As a result, we got a family of real quadratic fields with non-trivial class number.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

本研究の第１の学術的意義は, 部分商の最大値やその個数, 実二次体の類数, 分岐の様子など様々な性質を関連づけた点である. 連分数から導かれる情報は多種多様であるが, それらをそれぞれに意味づけし整理することは, 今後の研究にも不可欠である. 第２の意義は, ある条件を満たす代数体を明示的に与えた点である. 様々なケースにおいて扱いやすいものを具体的に与えることは, 学術的貢献に値すると考える.