Explicit representation of fundamental units of some quadratic fields
type: Item
instance of: scholarly article
title: Explicit representation of fundamental units of some quadratic fields
zbMATH Open document ID: 0831.11058
author: Koshi Tomita
published in: Proceedings of the Japan Academy. Series A
publication date: 1996-02-18T00:00:00Z
review text: It is well-known that for any given and fixed square-free positive integer \(D\) the fundamental unit of the real quadratic field \(\mathbb{Q} (\sqrt {D})\) is explicitly described by using the continued fraction theory. In this paper, the author intends to provide an explicit unified form of the fundamental units of some real quadratic fields. Namely, for any square-free positive integer \(d\) congruent to \(1\bmod 4\), denote by \(k_d\), \(\varepsilon_d= (T_d+ U_d \sqrt {d})> 1\) the period in the continued fraction expansion of \(\omega_d= (1+ \sqrt {D}) /2\), the fundamental unit of \(\mathbb{Q} (\sqrt {D})\) respectively and put \(d= a^2+ b\) \((0< b\leq 2a)\). Then, under the assumption of \(k_d =3\), the author proves the following: in the case of even \(a\), the continued fraction expansion of \(\omega_d\) is \([a/2, \overline {1, 1, a-1} ]\), and moreover \((T_d, U_d)= (2a,2)\) and \(d= a^2+ 1\) hold. In the case of odd \(a\), the continued fraction expansion of \(\omega_d\) is \([(a+ 1)/2, \overline {l, l, a}]\) for a positive integer \(l\), and moreover \((T_d, U_d)= ((l^2+ 1)^2 r+l ((l^2+ 3), l^2+1)\) and \(d= (l^2+1)^2 r^2+ 2l (l^2+ 3)r+ l^2+4\) hold for a positive integer \(r\) satisfying \(a= (l^2+ 1)r+1\).
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11Y65
zbMATH DE Number: 799701
zbMATH Keywords: fundamental units
zbMATH Keywords: real quadratic fields
zbMATH Keywords: continued fraction expansion
reviewed by: Hideo Yokoi
MaRDI profile type: Publication
full work available at URL: pjaa.71.41
OpenAlex ID: W1980779199
cites work: On the fundamental units and the class numbers of real quadratic fields
cites work: The fundamental unit and class number one problem of real quadratic fields with prime discriminant
cites work: The fundamental unit and bounds for class numbers of real quadratic fields
cites work: New invariants and class number problem in real quadratic fields
DOI: 10.3792/PJAA.71.41
description: scientific article; zbMATH DE number 799701