## Functiones et Approximatio Commentarii Mathematici

### 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 $3$

Akiko Ito

#### Abstract

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 [15]. 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 [4]. We add an application of this result to the Iwasawa invariants related to Greenberg's conjecture.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 49, Number 1 (2013), 111-135.

Dates
First available in Project Euclid: 20 September 2013

https://projecteuclid.org/euclid.facm/1379686438

Digital Object Identifier
doi:10.7169/facm/2013.49.1.8

Mathematical Reviews number (MathSciNet)
MR3127903

Zentralblatt MATH identifier
1273.11158

Subjects
Secondary: 11R29: Class numbers, class groups, discriminants

#### Citation

Ito, Akiko. 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 $3$. Funct. Approx. Comment. Math. 49 (2013), no. 1, 111--135. doi:10.7169/facm/2013.49.1.8. https://projecteuclid.org/euclid.facm/1379686438

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