Continued fractions and certain real quadratic fields of minimal type

• Kawamoto Fuminori

  Department of Mathematics, Faculty of Science, Gakushuin University

• Tomita Koshi

  Department of Mathematics, Meijo University

抄録

The main purpose of this article is to introduce the notion of real quadratic fields of minimal type in terms of continued fractions with period $¥ell$. We show that fundamental units of real quadratic fields that are not of minimal type are relatively small. So, we see by a theorem of Siegel that such fields have relatively large class numbers. Also, we show that there exist exactly 51 real quadratic fields of class number 1 that are not of minimal type, with one more possible exception. All such fields are listed in the table of Section 8.2. Therefore we study real quadratic fields with period $¥ell$ of minimal type in order to find real quadratic fields of class number 1, and first examine the case where $¥ell$ ≤ 4. In particular we obtain a result on Yokoi invariants m_d and class numbers h_d of real quadratic fields Q($¥sqrt{d}$) with period 4 of minimal type.

収録刊行物

• Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ

  Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ 60 (3), 865-903, 2008

  一般社団法人 日本数学会

キーワード

• continued fractions
• real quadratic fields
• fundamental units
• class numbers

詳細情報

• CRID: 1390001205114670336
• NII論文ID: 10024331965
• NII書誌ID: AA0070177X
• DOI: 10.2969/jmsj/06030865
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• NDL書誌ID: 9583082
• Web Site: http://id.ndl.go.jp/bib/9583082
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I9583082
• 本文言語コード: en
• データソース種別:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
• 抄録ライセンスフラグ: 使用不可