Abstract and Applied Analysis

Volume 2012, Article ID 570154, 4 pages

http://dx.doi.org/10.1155/2012/570154

## Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

Department of Mathematics, Faculty of Science, Istanbul University, 34134 Istanbul, Turkey

Received 17 October 2012; Accepted 4 November 2012

Academic Editor: Haydar Akca

Copyright © 2012 A. Pekin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2, where is an integer and the discriminant of such fields has only two prime divisors.

#### 1. Introduction

Let be the quadratic fields with discriminant and its class number. In the narrow sense, the class number of is denoted by , where, if , then and the fundamental unit has norm , otherwise . If the discriminant of has two distinct prime divisors, then by the genus theory of Gauss the 2-class group of is cyclic. The problem of the divisibility of class numbers for number fields has been studied by many authors. There are Hartung [1], Honda [2], Murty [3], Nagel [4], Soundararajan [5], Weinberger [6], Yamamoto [7], among them. Ankeny and Chowla [8] proved that there exists infinitely many imaginary quadratic fields each with class numbers divisible by where is any given rational integer. Later, Belabas and Fouvry [9] proved that there are infinitely many primes such that the class number of the real quadratic field is not divisible by 3. Furthermore, many authors [7, 10–13] have studied the conditions for to be divisible by when the 2-class group of is cyclic. However the criterion for to be divisible by is known for only and the existence of quadratic fields with arbitrarily large cyclic 2-class groups is not known yet. Recently, Byeon and Lee [14] proved that there are infinitely many imaginary quadratic fields whose ideal class group has an element of order and whose discriminant has only two prime divisors. In this paper, we will prove a theorem that the order of the ideal class group of certain imaginary quadratic field is divisible by . Moreover, we notice that the discriminant of these fields has only different two prime divisors. Finally, we will give a table as an application to our main theorem.

#### 2. Main Theorem

Our main theorem is the following.

Theorem 2.1. *Let be square-free integer with primes . If there is a prime satisfying , then for at least positive integer where .*

In order to prove this theorem we need the following fundamental lemma and some theorems.

Lemma 2.2. *If is of the form where and are primes , then there is a prime such that .*

*Proof. *Let and be quadratic nonresidues for and are primes such that , , where denotes Legendre symbol and . Therefore, by Chinese Remainder Theorem, we can write , for a positive integer . Now, we consider the numbers of the form such that for some . Since are distinct residues for some , then we get , . We assert that . Really, we suppose that , then there is a prime such that , and so we have , . Thereby this follows that , or . But since , then and ; this is in contradiction with , . Therefore, holds. Thus, by the Dirichlet theorem on primes, there is a prime satisfying . Hence, it is seen that .

The following theorem is generalized by Cowles [15].

Theorem 2.3. *Let , , be positive integers with and , and let be square-free and negative. If is not the norm of a primitive element of whenever properly divides , then .*

Cowles proved this theorem by using the decomposition of the prime divisors in . But Mollin has emphasized in [16] that it contains some misprints and then he has provided the following theorem which is more useful in practise than Theorem 2.4.

Theorem 2.4. *Let be a square-free integer of the form where , , and are positive integers such that and . If , then .*

Theorem 2.5. *Let be a square-free integer, and let , be integers such that*(i)* is the norm of a primitive element from ,*(ii)* is not the norm of a primitive element from for all properly dividing ,*(iii)*if , then .**Then divides the exponent of , where is the class group of .*

#### 3. Proof of Main Theorem

Now we will provide a proof for the fundamental theorem which is more practical than all of the works above mentioned.

*Proof. *From the assumption of Lemma 2.2, it follows that there is suitable prime with such that . However, from the properties of the Legendre symbol, we can write for any integer . Since , then we have . Therefore, there are integers such that the equation has a solution in integers. Hence, we can write , where . From this equation, it is seen that is the norm of a primitive element of , and, then by Theorem 2.5, divides .

We have the following results.

Corollary 3.1. *Let be a square-free and negative integer in the form of with , are positive integers and , , are primes such that , . If is the norm of a primitive element of , then the order of the ideal class group of is .*

Corollary 3.2. *Let be a square-free and negative integer in the form of , then there exists exactly 34433 imaginary quadratic fields satisfying assertion of the main theorem.*

#### 4. Table

The above-mentioned imaginary quadratic fields correspond to some values of which are given in Table 1. We have provided a table of the examples to illustrate the results above, using C programming language. Moreover, it is easily seen that the class numbers of imaginary quadratic fields of are divisible by from Table 1.

#### Acknowledgment

This work was partially supported by the scientific research project with the number IU-YADOP 12368.

#### References

- P. Hartung, “Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by $n$,”
*Journal of Number Theory*, vol. 6, pp. 276–278, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Honda, “A few remarks on class numbers of imaginary quadratic number fields,”
*Osaka Journal of Mathematics*, vol. 12, pp. 19–21, 1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. R. Murty, “The ABC conjecture and exponents of class groups of quadratic fields,” in
*Number Theory*, vol. 210 of*Contemporary Mathematics*, pp. 85–95, American Mathematical Society, Providence, RI, USA, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Nagel, “Über die Klassenzahl imaginär quadratischer Zahlkörper,”
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*, vol. 1, pp. 140–150, 1992. View at Google Scholar - K. Soundararajan, “Divisibility of class numbers of imaginary quadratic fields,”
*Journal of the London Mathematical Society. Second Series*, vol. 61, no. 3, pp. 681–690, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. J. Weinberger, “Real quadratic fields with class numbers divisible by
*n*,”*Journal of Number Theory*, vol. 5, pp. 237–241, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Yamamoto, “Divisibility by 16 of class number of quadratic fields whose 2-class groups are cyclic,”
*Osaka Journal of Mathematics*, vol. 21, no. 1, pp. 1–22, 1984. View at Google Scholar · View at MathSciNet - N. C. Ankeny and S. Chowla, “On the divisibility of the class number of quadratic fields,”
*Pacific Journal of Mathematics*, vol. 5, pp. 321–324, 1955. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Belabas and E. Fouvry, “Sur le 3-rang des corps quadratiques de discriminant premier ou presque premier,”
*Duke Mathematical Journal*, vol. 98, no. 2, pp. 217–268, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Barrucand and H. Cohn, “Note on primes of type ${x}^{2}+32{y}^{2}$, class number, and residuacity,”
*Journal für die Reine und Angewandte Mathematik*, vol. 238, pp. 67–70, 1969. View at Google Scholar · View at MathSciNet - H. Bauer, “Zur berechnung der 2-klassenzahl der quadratischen Zahlkörper mit genau zwei verschiedenen diskriminantenprimteilern,”
*Journal für die Reine und Angewandte Mathematik*, vol. 248, pp. 42–46, 1971. View at Google Scholar · View at MathSciNet - H. Hasse, “Über die teilbarkeit durch 2
^{3}der Klassenzahl imaginärquadratischer Zahlkörper mit genau zwei verschiedenen diskriminantenprimteilern,”*Journal für die Reine und Angewandte Mathematik*, vol. 241, pp. 1–6, 1970. View at Google Scholar · View at MathSciNet - P. Kaplan, K. S. Williams, and K. Hardy, “Divisibilité par 16 du nombre des classes au sens strict des corps quadratiques réels dont le deux-groupe des classes est cyclique,”
*Osaka Journal of Mathematics*, vol. 23, no. 2, pp. 479–489, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Byeon and S. Lee, “Divisibility of class numbers of imaginary quadratic fields whose discriminant has only two prime factors,”
*Japan Academy. Proceedings. Series A. Mathematical Sciences*, vol. 84, no. 1, pp. 8–10, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. J. Cowles, “On the divisibility of the class number of imaginary quadratic fields,”
*Journal of Number Theory*, vol. 12, no. 1, pp. 113–115, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. A. Mollin, “Diophantine equations and class numbers,”
*Journal of Number Theory*, vol. 24, no. 1, pp. 7–19, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet