A Note on the Structure of Certain Real Quadratic Number Fields
Iranian Journal of Science and Technology, Transaction A: Science, cilt.41, sa.3, ss.759-769, 2017 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 41 Sayı: 3
- Basım Tarihi: 2017
- Doi Numarası: 10.1007/s40995-017-0292-z
- Dergi Adı: Iranian Journal of Science and Technology, Transaction A: Science
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Sayfa Sayıları: ss.759-769
- Anahtar Kelimeler: Continued fraction, Fundamental unit, Class number and quadratic forms
- Kırklareli Üniversitesi Adresli: Evet
Özet
The aim of this paper is to determine the general forms of the continued fraction expansions of the quadratic irrational number wd which is integral basis element of Z[1+d2], also determine td,ud which are the coefficients of fundamental units εd=(td+udd)/2⟩1 of the real quadratic number fields Q(d) using a new explicit formula. Fundamental units are calculated with this algorithm in an easy way for the period kd which is equal to 9 in the continued fraction expansion of wd for such real quadratic fields where d≡1(mod4) is a positive square free integer. Moreover, some results are given on Yokoi’s invariant value nd which is defined in the terms of coefficients of fundamental unit and the class number hd of the real quadratic number field Q(d) as well as reduced indefinite quadratic forms fd.