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

Authors: A. Pekin
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2g, where g>1 is an integer and the discriminant of such fields has only two prime divisors.


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## Body

## 1. Introduction

LetK=Q(D) be the quadratic fields with discriminant D and h=h(D) its class number. In the narrow sense, the class number of K is denoted by h+(D), where, if D>0, then h+(D)=2h(D) and the fundamental unit εD has norm 1, otherwise h+(D)=h(D). If the discriminant of |D| has two distinct prime divisors, then by the genus theory of Gauss the 2-class group of K 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 g where g is any given rational integer. Later, Belabas and Fouvry [9] proved that there are infinitely many primes p such that the class number of the real quadratic field K=Q(p) is not divisible by 3. Furthermore, many authors [7, 10–13] have studied the conditions for h+(D) to be divisible by 2n when the 2-class group of K is cyclic. However the criterion for h+(D) to be divisible by 2n is known for only n≤4 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 2g 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 2g. 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.
LetD=pq be square-free integer with primes p≡q≡1(mod4). If there is a prime r≡1(mod8) satisfying (D/r)=1, then t∣h(D) for at least positive integer t where t≥2.In order to prove this theorem we need the following fundamental lemma and some theorems.Lemma 2.2.
IfD is of the form p·q where p and q are primes p≡q≡1(mod4), then there is a prime r≡1(mod8) such that (D/r)=1.Proof.
Leta and b be quadratic nonresidues for p and q are primes such that (a/p)=-1, (b/q)=-1, where () denotes Legendre symbol and g·c·d(p,q)=1. Therefore, by Chinese Remainder Theorem, we can write w≡a(modp), w≡b(modq) for a positive integer w. Now, we consider the numbers of the form pqk+w such that pqk0+w≡1(mod8) for some 1≤k0≤8. Since pqk0+w are distinct residues mod(8) for some 1≤k0≤8, then we get pq(8n+k0)+w=8pqn+pqk0+w, n≥0. We assert that g·c·d(8pq,pqk0+w)=1. Really, we suppose that g·c·d(8pq,pqk0+w)=m>1, then there is a prime s such that s∣m, and so we have s∣8pq, s∣pqk0+w. Thereby this follows that s=2, p or q. But since pqk0+w≡1(mod8), then s≠2 and s∣m; this is in contradiction with w≡a(modp), w≡b(modq). Therefore, g·c·d(8pq,pqk0+w)=1 holds. Thus, by the Dirichlet theorem on primes, there is a prime r satisfying r=pq(8n+k0)+w=8pqn+pqk0+w. Hence, it is seen that r≡1(mod8).The following theorem is generalized by Cowles [15].Theorem 2.3.
Letr, m, t be positive integers with m>1 and t>1, and let n=r2-4mt be square-free and negative. If mc is not the norm of a primitive element of OK whenever c properly divides t, then t∣h(n).Cowles proved this theorem by using the decomposition of the prime divisors inOK. 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.
Letn be a square-free integer of the form n=r2-4mt where r, m, and t are positive integers such that m>1 and t>1. If r2≤4mt-1(m-1), then t∣h(n).Theorem 2.5.
Letn be a square-free integer, and let m>1, t>1 be integers such that(i)
∓mt is the norm of a primitive element from K=Q(n),(ii)
∓mc is not the norm of a primitive element from K for all c properly dividing t,(iii)
ift=|m|2, then n≡1(mod8).
Thent divides the exponent of ψK, where ψK is the class group of K.


## 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 Lemma2.2, it follows that there is suitable prime r with r≡1(mod8) such that (D/r)=1. However, from the properties of the Legendre symbol, we can write (Dy2/r2)=1 for any integer y. Since (2,r)=1, then we have (Dy2/rt)=1. Therefore, there are integers x=a/2,y=b/2 such that the equation x2-Dy2=∓rt has a solution in integers. Hence, we can write a2-Db2=∓4rt, where a≡b(mod2). From this equation, it is seen that rt is the norm of a primitive element of OK, and, then by Theorem 2.5, t divides h(n).We have the following results.Corollary 3.1.
LetD be a square-free and negative integer in the form of D=n2-4r2g=p·q with n>1, g>1 are positive integers and p, q, r are primes such that p≡q≡1(mod4), r≡1(mod8). If r2g is the norm of a primitive element of OK, then the order of the ideal class group of K=Q(D) is 2g.Corollary 3.2.
LetD be a square-free and negative integer in the form of D=p·q, then there exists exactly 34433 imaginary quadratic fields satisfying assertion of the main theorem.


## 4. Table

The above-mentioned imaginary quadratic fieldsK=Q(D) correspond to some values of D(5≤D≤106) 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 K=(QD) are divisible by 2g from Table 1.Table 1
D
p
q
r
h
(
D
)
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



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*Source: 570154-2012-12-09.xml*