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Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors
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.
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 , Honda , Murty , Nagel , Soundararajan , Weinberger , Yamamoto , among them. Ankeny and Chowla  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  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  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 .
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  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.
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.
This work was partially supported by the scientific research project with the number IU-YADOP 12368.
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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.