Fundamental unit system and class number of real bicyclic biquadratic number fields
type: Item
instance of: scholarly article
title: Fundamental unit system and class number of real bicyclic biquadratic number fields
zbMATH Open document ID: 1011.11071
published in: Proceedings of the Japan Academy. Series A
publication date: 2003-06-04T00:00:00Z
review text: For three certain types of real bicyclic biquadratic number fields, the author intends to determine explicitly the class numbers and the fundamental systems of the unit groups by using the class numbers and the fundamental units of their quadratic subfields. Namely, let \(D_1= n^2+2\), \(D_2= n^4+2n^2+2\), \((2\mid n\geq 4)\), and let \(D_1,D_2\) be square free. Then the class number \(h_K\) and the fundamental system \(U_K\) of the unit groups of \(K= \mathbb{Q}(\sqrt{n^2+2}, \sqrt{n^4+2n^2+2})\) are \(h_K= (\frac 12)h_1h_2h_3\) and \(U_K= \{-1, \varepsilon_1, \varepsilon_2, \sqrt{\varepsilon_1 \varepsilon_3}\}\), where \(h_i\) and \(\varepsilon_i\) mean respectively the class number and the fundamental unit of subfields \(k_i= \mathbb{Q}(\sqrt{D_i})\) for \(i=1,2\), and \(k_3= \mathbb{Q} (\sqrt{D_1D_2})\). The other remainder types are the following: \(D_1= (n^2+rd) (n^2+sd)\), \(D_2= (n^2+td) (n^2+ud)\) and \(D_1= n^2+d\), \(D_2= m^2n^2+r\).
reviewed by: Hideo Yokoi
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1779508
zbMATH Keywords: real bicyclic biquadratic number fields
zbMATH Keywords: class numbers
zbMATH Keywords: fundamental units
author: Kun Peng Wang
MaRDI profile type: Publication
full work available at URL: pjaa.77.147
OpenAlex ID: W2035223336
cites work: Fundamental systems of units in number fields \(Q(\sqrt{D^ 2+d},\sqrt{D^ 2+4d})\) with \(d|D\)
cites work: Fundamental systems of units in biquadratic parametric number fields
cites work: Q5541475
cites work: Fundamental unit system and class number for real number field of type \((2,2,2)\).
cites work: Q4724723
DOI: 10.3792/PJAA.77.147
description: scientific article; zbMATH DE number 1779508