On the divisibility of class numbers of imaginary quadratic fields \((\mathbb{Q}(\sqrt{D}),\mathbb{Q}(\sqrt{D+m}))\)
\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 \((x,z)\), \((u,v)\).
scientific article; zbMATH DE number 7345284
scientific article