On the divisibility of class numbers of imaginary quadratic fields \((\mathbb{Q}(\sqrt{D}),\mathbb{Q}(\sqrt{D+m}))\)
type: Item
instance of: scholarly article
title: On the divisibility of class numbers of imaginary quadratic fields \((\mathbb{Q}(\sqrt{D}),\mathbb{Q}(\sqrt{D+m}))\)
zbMATH Open document ID: 1471.11276
author: Jian-Feng Xie
author: Kuok Fai Chao
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: \textit{Y. Iizuka} recently proved that there exist infinitely many imaginary quadratic fields \(\mathbb Q(\sqrt{D})\) and \(\mathbb Q(\sqrt{D+1})\) both of whose class numbers are divisible by 3 [J. Number Theory 184, 122--127 (2018; Zbl 1420.11138)]. The main result of the paper under review is that for any odd positive integer \(n\) and any positive integer \(m\), there exist infinitely many pairs of imaginary quadratic fields \(\mathbb Q(\sqrt{D})\) and \(\mathbb Q(\sqrt{D+m})\) both of whose ideal class groups have an element of order \(n\). The construction of these pairs is based on an extension of a result of \textit{Y. Yamamoto} [Osaka J. Math. 7, 57--76 (1970; Zbl 0222.12003)], who gave necessary conditions for the existence of an element of order \(n\) in a quadratic field of the type \(\mathbb Q(\sqrt{x^2-4z^n})\), where \(x,y,z\) are integers giving a solution to the Diophantine equation \(X^2-Y^2D=4Z^n\) and \(x,z\) are relatively prime integers satisfying several additional conditions. Here, an analogous result is obtained for quadratic fields \(\mathbb Q(\sqrt{x^{2m}-4x^kz^n})\), where \(m,k,n\) are given positive integers and \(x,y,z\) are now integers giving a solution to the Diophantine equation \(X^{2m}-Y^2D=4X^kZ^n\) and satisfying appropriate additional conditions. The main part of the proof of the theorem then consists of showing that there exist integer pairs \((x,z)\) and \((u,v)\) satisfying the required conditions and for which \[ x^2-4z^n = a^2(u^2-4v^n-m) \] for some \(a\in \mathbb Z\). Then the imaginary quadratic fields \(\mathbb Q(\sqrt{u^2-4v^n})\) and \(\mathbb Q(\sqrt{x^2-4z^n})\) have class groups with the desired property. Finally, it is shown that there exist infinitely many suitable pairs \((x,z)\), \((u,v)\).
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 7345284
zbMATH Keywords: quadratic field
zbMATH Keywords: class number
zbMATH Keywords: ideal class group
MaRDI profile type: Publication
full work available at URL: s11139-019-00217-1
OpenAlex ID: W3011572152
cites work: On the class number divisibility of pairs of imaginary quadratic fields
cites work: On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves
cites work: An infinite family of pairs of quadratic fields Q(\sqrt D) and Q(\sqrt mD) whose class numbers are both divisible by 3
cites work: An infinite family of pairs of imaginary quadratic fields with ideal classes of a given order
cites work: On unramified Galois extensions of quadratic number fields
DOI: 10.1007/S11139-019-00217-1
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description: scientific article; zbMATH DE number 7345284