Calculation of the class numbers and fundamental units of abelian extensions over imaginary quadratic fields from approximate values of elliptic units
type: Item
instance of: scholarly article
title: Calculation of the class numbers and fundamental units of abelian extensions over imaginary quadratic fields from approximate values of elliptic units
zbMATH Open document ID: 0591.12010
author: Ken Nakamula
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: Let F be \({\mathbb{Q}}\) or an imaginary quadratic field, and let L be an abelian extension of F (with the condition L real if \(F={\mathbb{Q}})\). The author gives a general procedure to compute the class number h of L and finds simultaneously the fundamental units of L. When \(F={\mathbb{Q}}\), the principle of calculation has been introduced by \textit{G. Gras} and the reviewer [Bull. Sci. Math., II. Sér. 101, 97-129 (1977; Zbl 0359.12007)]. When F is an imaginary quadratic field, the author uses the elliptic units instead of the cyclotomic ones, and gives an explicit procedure for a computer. As in the absolute abelian case, the method consists in computing an explicit upper bound for the index of elliptic units in units of L, and in deducing this index from an approximate value of the elliptic units. The author gives some numerical examples, and has applied this algorithm particularly in the cubic, quartic and sextic cases [Proc. Japan Acad., Ser. A 57, 56-59, 117-120, 363-366 (1981; Zbl 0478.12003, Zbl 0478.12004, Zbl 0515.12003)].
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3948416
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: abelian extension
zbMATH Keywords: class number
zbMATH Keywords: fundamental units
zbMATH Keywords: elliptic units
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03720245
OpenAlex ID: W2066352645
DOI: 10.2969/JMSJ/03720245
description: scientific article; zbMATH DE number 3948416