The main purpose of this article is to introduce the notion of real quadratic fields of minimal type in terms of continued fractions with period $\ell$ . We show that fundamental units of real quadratic fields that are not of minimal type are relatively small. So, we see by a theorem of Siegel that such fields have relatively large class numbers. Also, we show that there exist exactly $51$ real quadratic fields of class number $1$ that are not of minimal type, with one more possible exception. All such fields are listed in the table of Section 8.2. Therefore we study real quadratic fields with period $\ell$ of minimal type in order to find real quadratic fields of class number $1$ , and first examine the case where $\ell\le 4$ . In particular we obtain a result on Yokoi invariants $m_d$ and class numbers $h_d$ of real quadratic fields $\bm{Q}(\sqrt{d})$ with period $4$ of minimal type.
Publié le : 2008-07-15
Classification:
continued fractions,
real quadratic fields,
fundamental units,
class numbers,
11R29,
11A55,
11R11,
11R27
@article{1217884495,
author = {KAWAMOTO, Fuminori and TOMITA, Koshi},
title = {Continued fractions and certain real quadratic fields of minimal type},
journal = {J. Math. Soc. Japan},
volume = {60},
number = {1},
year = {2008},
pages = { 865-903},
language = {en},
url = {http://dml.mathdoc.fr/item/1217884495}
}
KAWAMOTO, Fuminori; TOMITA, Koshi. Continued fractions and certain real quadratic fields of minimal type. J. Math. Soc. Japan, Tome 60 (2008) no. 1, pp. 865-903. http://gdmltest.u-ga.fr/item/1217884495/