On the divisibility of class numbers of imaginary quadratic fields ( $$ {\mathbb {Q}}(\sqrt{D}), {\mathbb {Q}}(\sqrt{D+m
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On the divisibility of √ √ class numbers of imaginary quadratic fields (Q( D), Q( D + m)) Jian-Feng Xie1 · Kuok Fai Chao2 Received: 3 December 2018 / Accepted: 26 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract For a given odd positive number n and a positive integer√m, we show√ that there exist infinitely many pairs of imaginary quadratic fields Q( D) and Q( D + m) with D ∈ Z whose class groups have an element of order n. Keywords Quadratic field · Class number · Ideal class group Mathematics Subject Classification 11R29 · 11R11
1 Introduction The study of class numbers of algebraic number fields is one of the main topics in number theory. In general, the class number is not easy to be computed. Instead we focus on the divisibility property of the class number. In other words, we try to figure out some factors of the order of class group. Here we review some results which are related √ to this paper. √ In [3], Komatsu gives an infinite family of pairs of quadratic fields Q( D) and Q( m D) with m, D ∈ Z whose class numbers are both divisible √ by 3. D) and In [2], the authors construct an infinite family of pairs of quadratic fields Q( √ Q( m D + n) with D ∈ Q, m, n ∈ Z whose class numbers are both divisible by 3 or 5 or 7. Komatsu’s remarkable result √ [4] gives an √ √ infinite family of imaginary quadratic fields Q( D) such that both Q( D) and Q( m D) have ideal classes of order n, in which m and n are natural numbers greater than one. Recently, Iizuka √ √ [1] gives an infinite family of pairs of imaginary quadratic fields Q( D) and Q( D + 1) with
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Kuok Fai Chao [email protected] Jian-Feng Xie [email protected]
1
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, China
2
Department of Mathematic, College of Science, Shanghai University, Shanghai, China
123
J.-F. Xie, K. F. Chao
D ∈ Z whose class numbers are divisible by 3, and proposes the following conjecture: Conjecture 1.1 For any prime number p and any positive integer n, there is an infinite family of n + 1 successive real (or imaginary) quadratic fields Q
√ √ √ D ,Q D + 1 ,...,Q D+n
with D ∈ Z whose class numbers are divisible by p. In this paper, we study some special cases on this conjecture. Indeed, we show that Theorem 1.2 For any odd positive integer√n and any √ positive integer m, there are infinitely many pairs of imaginary fields Q( D) and Q( D + m) whose class groups have an element of order n respectively. The theorem above-mentioned can be viewed as a generalization of Conjecture 1.1. For example, the class numbers of the pair Q( −5,078,303,128,661,121), Q( −5,078,303,128,661,120) are 8,239,520 and 7,731,840 respectively. They can be divided by 5. The construction of these pairs of quadratic fields is√based on the study of the ideal class group of quadratic fields in the form of Q( x 2m − 4x k z n ) where x, z are variables and m, k, n are positive integers. These details are in Sects. 2 and 3. In Sect. 4, we discuss the local-gl
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