Aoki, Miho; Kishi, Yasuhiro A family of pairs of imaginary cyclic fields of degree \((p-1)/2\) with both class numbers divisible by \(p\). (English) Zbl 1447.11113 Ramanujan J. 52, No. 1, 133-161 (2020). In their previous paper [J. Number Theory 176, 333–343 (2017; Zbl 1422.11221)] the authors of the paper under review proved the following theorem where \(F_n\) is the \(n\)-th Fibonacci number.Theorem 1. For \(n\in {\mathcal N} := \{n\in {\mathbb N} \mid n\equiv \pm 3\pmod{500}, n\not\equiv 0 \pmod 3\}\), the class numbers of both \({\mathbb Q}(\sqrt{2-F_n})\) and \({\mathbb Q}(\sqrt{5(2-F_n)})\) are divisible by \(5\). Moreover, the set of pairs \[ \left \{({\mathbb Q}(\sqrt{2-F_n}), {\mathbb Q}(\sqrt{5(2-F_n)}))\Big |n\in {\mathcal N}\right\} \] is infinite.In the present paper they obtain a result of which Theorem 1 is a special case. To state this result let \(p\) be a prime number such that \(p\equiv 5 \pmod 8\) and let \(\zeta\) be a primitive \(p\)-th root of unity. Let \(\delta\) be a generator of \(\mathrm{Gal}({\mathbb Q} (\zeta)/{\mathbb Q})\) and put \(\delta_0 := \delta^{(p-1)/4}\). Let \(\omega_0 := \zeta + \zeta^{\delta_0} + \zeta^{\delta_0^2} + \zeta^{\delta_0^3}\). Then \(k_0 := {\mathbb Q}(\omega_0)\) is the unique subfield of \({\mathbb Q} (\zeta)\) of degree \((p-1)/4\). Let \(u_p > 1\) be the fundamental unit of \(k={\mathbb Q}(\sqrt{p})\). Then \[ u_p=\frac{t+b\sqrt{p}}{2} \] where \(t,b\in {\mathbb Z}\) with \(t, b >0\). Now, using the trace \(t\) of \(u_p\), define the following two sequences \(\{{\mathcal F}_n\}\) and \(\{{\mathcal L}_n\}\) by \[ \begin{cases} {\mathcal F}_0 := 0,\ {\mathcal F}_1 := 1, \ {\mathcal F}_{n+2} := t{\mathcal F}_{n+1} +{\mathcal F}_n \ (n\in {\mathbb Z}),\\ \\ {\mathcal L}_0 := 2,\ {\mathcal L}_1 := t, \ {\mathcal L}_{n+2} := t{\mathcal L}_{n+1} +{\mathcal L}_n \ (n\in {\mathbb Z}). \end{cases} \] For integers \(m\) and \(n\), and a prime number \(q\) \((\neq p)\), put \[ D_{m,n} := {\mathcal L}_m(2{\mathcal F}_m - {\mathcal F}_n{\mathcal L}_m)b, \] \[ N_q := \begin{cases} \mathrm{lcm}(p^2(p-1), q-1) \text{ if } \left (\frac{p}{q}\right )=1,\\ \mathrm{lcm}(p^2(p-1), 2(q+1))\text{ if } \left (\frac{p}{q}\right)=-1. \end{cases} \] (As the authors point out, when \(m\) and \(n\) are odd and \(n>3\), \(D_{m,n}\) is negative since \({\mathcal F}_{-m} = (-1)^{m+1}{\mathcal F}_m\) and \({\mathcal L}_{-m} = (-1)^m{\mathcal L}_m\)). Finally, put \[ \alpha = \alpha(m,n) := \frac{{\mathcal L}_n {\mathcal L}_m + ({\mathcal L}_m{\mathcal F}_n - 2{\mathcal F}_m)b\sqrt{p}}{2}, \] \[ f_{\alpha}(X) := X^4 -TX^3+(N+2)X^2 - TX + 1, \] \[ f_{\alpha, q}(X) := f_{\alpha} \bmod q \in {\mathbb F}_q[X], \] where \(N := N_{k/{\mathbb Q}}(\alpha)\) and \(T :=\mathrm{Tr}_{k/{\mathbb Q}}(\alpha)\). The first main result the authors obtain isMain Theorem 1. We assume that there exist integers \(m_0,n_0\) with \(m_0\equiv n_0 \equiv 1 \pmod 2\) and a prime number \(q\) such that (i) \(({\mathcal L}_{m_0}{\mathcal F}_{n_0} - 2{\mathcal F}_{m_0})b \equiv 0 \pmod {p^2}\),(ii) \(q \nmid 2bp\) and \(f_{\alpha_0,q}(a) = 0\) for some \(i\in \{1,2,4\}\) and \(a\in {\mathbb F}_{q^i}\setminus {\mathbb F}^p_{q^i}\), where \(\alpha_0 := \alpha(m_0,n_0)\). Then for any pair \((m,n)\in {\mathcal N} := \left \{(m,n)\in {\mathbb Z}^2 \Big | m\equiv m_0 \pmod {N_q}, n\equiv n_0 \pmod{N_q}, n>3 \right \}\), the class numbers of both imaginary cyclic fields \(k_0(\sqrt{D_{m,n}})\) and \(k_0(\sqrt{pD_{m,n}})\) of degre \((p-1)/2\) are divisible by \(p\). Moreover, the set of pairs \[ \left \{(k_0(\sqrt{D_{m,n}}), k_0(\sqrt{pD_{m,n}})) \Big |(m,n)\in {\mathcal N}\right \} \] is infinite.The second main result of the paper isMain Theorem 2. Assume that ERH holds. Then there exist the integers \(m_0, n_0\) and the prime number \(q\) as in Main Theorem 1.Here “ERH” means the extended Riemann hypothesis for \(k(\zeta_n, \sqrt [n]{u_p})\) for every square free integer \(n>0\).The authors conclude their paper with some examples illustrating Main Theorem 1. One of these shows how Main Theorem 1 implies Theorem 1. Reviewer: James E. Carter (Charleston) MSC: 11R11 Quadratic extensions 11R16 Cubic and quartic extensions 11R29 Class numbers, class groups, discriminants Keywords:class numbers; abelian number fields; fundamental units; Gauss sums; Jacobi sums; linear recurrence sequences Citations:Zbl 1422.11221 PDFBibTeX XMLCite \textit{M. Aoki} and \textit{Y. Kishi}, Ramanujan J. 52, No. 1, 133--161 (2020; Zbl 1447.11113) Full Text: DOI arXiv References: [1] Alaca, S.; Williams, KS, Introductory Algebraic Number Theory (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1035.11001 [2] Aoki, M.; Kishi, Y., On systems of fundamental units of certain quartic fields, Int. J. Number Theory, 11, 7, 2019-2035 (2015) · Zbl 1331.11099 [3] Aoki, M.; Kishi, Y., An infinite family of pairs of imaginary quadratic fields with both class numbers divisible by five, J. Number Theory, 176, 333-343 (2017) · Zbl 1422.11221 [4] Berndt, BC; Evans, RJ; Williams, KS, Gauss and Jacobi Sums (1998), New York: Wiley, New York [5] Iizuka, Y.; Konomi, Y.; Nakano, S., On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves, J. Math. Soc. Jpn., 68, 899-915 (2016) · Zbl 1354.11067 [6] Imaoka, M.; Kishi, Y., On dihedral extensions and Frobenius extensions: Galois theory and modular forms, Dev. Math., 11, 195-220 (2004) · Zbl 1068.11070 [7] Komatsu, T., An infinite family of pairs of quadratic fields \({\mathbb{Q}}(\sqrt{D})\) and \({\mathbb{Q}}(\sqrt{mD})\) whose class numbers are both divisible by \(3\), Acta Arith., 104, 129-136 (2002) · Zbl 1004.11062 [8] Komatsu, T., An infinite family of pairs of imaginary quadratic fields with ideal classes of a given order, Int. J. Number Theory, 13, 2, 253-260 (2017) · Zbl 1370.11125 [9] Lenstra, HW Jr, On Artin’s conjecture and Euclid’s algorithm in global fields, Invent. Math., 42, 201-224 (1977) · Zbl 0362.12012 [10] Ribenboim, P., The New Book of Prime Number Records (1996), New York: Springer, New York · Zbl 0856.11001 [11] Scholz, A., Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. Reine Angew. Math., 166, 201-203 (1932) · Zbl 0004.05104 [12] Takagi, T., Elementary Number Theory Lecture (1971), Tokyo: Kyoritsu Shuppan, Tokyo [13] Washington, LC, Introduction to Cyclotomic Fields (1982), New York: Springer, New York · Zbl 0484.12001 [14] Weil, A., Sur les courbes algébriques et les variétés quis’en déduisent (1948), Paris: Hermann, Paris · Zbl 0036.16001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.