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Existence of an infinite family of pairs of quadratic fields \(\mathbb{Q}(\sqrt{m_1D})\) and \(\mathbb{Q}(\sqrt{m_2D})\) whose class numbers are both divisible by 3 or both indivisible by 3. (English) Zbl 1273.11158

Summary: Let \(m_1, m_2\), and \(m_3\) be distinct square-free integers (including 1). First, we show that there exist infinitely many square-free integers \(d\) with \(\gcd(m_1m_2, d) = 1\) such that the class numbers of \(\mathbb{Q}(\sqrt{m_1d})\) and \(\mathbb{Q}(\sqrt{m_2d})\) are both divisible by 3. This is a generalization of a result of T. Komatsu [Acta Arith. 104, No. 2, 129–136 (2002; Zbl 1004.11062)]. Secondly, we show that there exist infinitely many positive fundamental discriminants \(D\) with \(\gcd(m_1m_2m_3, D) = 1\) such that the class numbers of real quadratic fields \(\mathbb{Q}(\sqrt{m_1D}), \mathbb{Q}(\sqrt{m_2D})\), and \(\mathbb{Q}(\sqrt{m_3D})\) are all indivisible by 3 when \(m_1, m_2\), and \(m_3\) are positive. This is a generalization of a result of D. Byeon [Proc. Am. Math. Soc. 132, No. 11, 3137–3140 (2004; Zbl 1061.11054)]. We add an application of this result to the Iwasawa invariants related to Greenberg’s conjecture.

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
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References:

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