A note on the fundamental unit in some types of the real quadratic number fields
8th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences, AMiTaNS 2016, Albena, Bulgaristan, 22 - 27 Haziran 2016, cilt.1773, (Tam Metin Bildiri)
- Yayın Türü: Bildiri / Tam Metin Bildiri
- Cilt numarası: 1773
- Doi Numarası: 10.1063/1.4964974
- Basıldığı Şehir: Albena
- Basıldığı Ülke: Bulgaristan
- Anahtar Kelimeler: Quadratic field, continued fraction, fundamental unit, special sequence
- Kırklareli Üniversitesi Adresli: Evet
Özet
Let k=Q(√d) be a real quadratic numbefield where d > 0 is a positive square-free integer. The map d→Q(√d) is a bijection from the set off all square-free integers d ≠ 0, 1 to the set of all quadratic fields Q(√d)={x+y√d|x,-rfpy1∈Q}. Furthermore, integral basis element of algebraic integer's ring in real quadratic fields is determined by either wd=√d= [a0;a1,a2,⋯,aℓ(d)-1,2a0] in the case of d ≡ 2,3(mod 4) or wd=1+√d/2= [a0;a1,a2,⋯,aℓ(d)-1,2a0-1] in the case of d ≡ 1(mod 4) where ℓ(d) is the period length of continued fraction expansion. The purpose of this paper is to obtain classification of some types of real quadratic fields Q(√d), which include the specific form of continued fraction expansion of integral basis element wd, for which has all partial quotient elements are equal to each other and written as ξs (except the last digit of the period) for ξ positive even integer where period length is ℓ=ℓ(d) and d ≡ 2,3(mod 4) is a square free positive integer. Moreover, the present paper deals with determining new certain parametric formula of fundamental unit ϵd=td+ud√d/2 > 1 with norm N(ϵd)=(-1)ℓ(d) for such types of real quadratic fields. Besides, Yokoi's d-invariants nd and md in the relation to continued fraction expansion of wd are calculated by using coefficients of fundamental unit. All supported results are given in numerical tables. These new results and tables are not known in the literature of real quadratic fields.