A Note on the Structure of Certain Real Quadratic Number Fields

Abstract

The aim of this paper is to determine the general forms of the continued fraction expansions of the quadratic irrational number \(w_{d} \,\) which is integral basis element of \(Z\left[ {\frac{1 + \sqrt d }{2}} \right]\), also determine \(t_{d} \,,\,\,u_{d} \,\) which are the coefficients of fundamental units \(\varepsilon_{d} \, = \,{{\left( {\,t_{d} \, + \,u_{d} \,\sqrt {\,d\,} } \right)} \mathord{\left/ {\vphantom {{\left( {\,t_{d} \, + \,u_{d} \,\sqrt {\,d\,} } \right)} 2}} \right. \kern-0pt} 2}\,\rangle \,1\) of the real quadratic number fields \({\mathbb{Q}}\left( {\sqrt d } \right)\) using a new explicit formula. Fundamental units are calculated with this algorithm in an easy way for the period \(k_{d} \,\) which is equal to 9 in the continued fraction expansion of \(w_{d} \,\) for such real quadratic fields where \(d \equiv 1\left( {\bmod 4} \right)\) is a positive square free integer. Moreover, some results are given on Yokoi’s invariant value \(n_{d} \,\) which is defined in the terms of coefficients of fundamental unit and the class number \(h_{d}\) of the real quadratic number field \({\mathbb{Q}}\left( {\sqrt d } \right)\) as well as reduced indefinite quadratic forms \(f_{d}\).