Abstract and Applied Analysis

Volume 2012 (2012), Article ID 570154, 4 pages

doi:10.1155/2012/570154

Research Article

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

A. Pekin

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.

Table 1

65
5
13
17
8

1165
5
233
41
20

3341
13
257
41
72

10685
5
2137
73
116

30769
29
1061
41
112

45349
101
449
17
168

95509
149
641
17
176

97309
73
1333
89
216

102689
29
3541
73
496

125009
41
3049
17
504

18497
53
349
41
168

20453
113
181
17
116

223721
137
1633
97
496

378905
5
75781
41
592

567137
17
333613
89
640

650117
13
50009
17
848

735929
373
1973
41
1664

847085
5
169417
73
936

874589
241
3629
17
1160

875705
5
175141
41
1328

876461
53
16537
73
1584

971081
109
8909
17
1464

971413
29
33497
73
336

978809
13
75293
89
1728

987169
97
10177
17
624

999997
757
1321
17
380

Acknowledgment

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

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