A Note on the Fundamental Unit in Some Types of the Real Quadratic Number Fields

Abstract

Let k = Q(root d) be a real quadratic numbefield where d > 0 is a positive square-free integer. The map d -> Q(root d) is a bijection from the set off all square-free integers d # 0,1 to the set of all quadratic fields Q(root d) = {x + y root d vertical bar x, y is an element of Q}. Furthermore, integral basis element of algebraic integer's ring in real quadratic fields is determined by either w(d) = root d = [a(0); a(1), a(2), ..., a(l(d)) (-) (1), 2a(0) in the case of d equivalent to 2,3 (mod 4) or w(d) = 1+root d/2 = [a(0), a(1), a(2), ..., a(l) ((d) -1) 2a(0) -1 in the case of d equivalent to 1(mod 4) where f (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(root d), which include the specific form of continued fraction expansion of integral basis element w(d), for which has all partial quotient elements are equal to each other and written as xi s (except the last digit of the period) for xi positive even integer where period length is l = l(d) and d equivalent to 2,3 (mod 4) is a square free positive integer. Moreover, the present paper deals with determining new certain parametric formula of fundamental unit epsilon(d) = t(d)+u(d)root d > 1 with norm N(epsilon(d)) = (-1)(l(d)) for such types of real quadratic fields. Besides, Yokoi's d-invariants n(d) and m(d) in the relation to continued fraction expansion of w(d) 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.