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.

1. Introduction

Let K=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.

Let D=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.

If D 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.

Let a 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.

Let r, 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 in OK. 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 n 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.

Let n be a square-free integer, and let m>1, t>1 be integers such that

∓mt is the norm of a primitive element from K=Q(n),

∓mc is not the norm of a primitive element from K for all c properly dividing t,

if t=|m|2, then n≡1(mod8).

Then t 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 Lemma 2.2, 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.

Let D 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.

Let D 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 fields K=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.

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

Acknowledgment

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

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