A Necessary and Sufficient Condition for to Have Class Number

Abstract

In this paper, we give a necessary and sufficient condition for real quadratic field to have class number where and is the number of odd prime divisors of .

1. Introduction

Let be a positive square-free integer and let and denote the class number and the class group of a real quadratic field , respectively.

The class number problem of quadratic fields is one of the most intriguing unsolved problems in algebraic number theory, and it has been the object of attention for many years of research.

In [1], Gauss had conjectured that there exist exactly nine imaginary quadratic fields of class number 1. Later, this was solved after diverse works of Stark, Heegner, and Baker. On the other hand, it was conjectured by Gauss that there are infinitely many real quadratic fields with class number one. This conjecture is still open.

Many fruitful research studies have been conducted in this direction (see [2–7]). In particular, we mention some results showing finiteness of class number one real quadratic fields in special families of real quadratic fields. Biro (see [2, 3])proved in 2003 two important results: Yokoi’s conjecture which asserts that only for six values of and Chowla’s conjecture which says that only for six values of .

Another line of work, in these contexts, is finding bounds for class number of a number field. Hasse [8] and Yokoi [9, 10] studied lower bounds for class numbers of certain real quadratic fields. Mollin [11, 12] generalized their results for certain real quadratic and biquadratic fields. Recently, Chakraborty et al. [13] derived a lower bound for . Also, Mishra [14] gave a lower bound for , where . For a given fixed number , it is interesting to find necessary and sufficient conditions for a real quadratic field to have class number , see [15–18]. In particular, Yokoi [18] showed that the class number of is 1 if and only if (with ) is a prime. In this work, we use the lower bound for to give a necessary and sufficient condition for to have class number where .

2. Preliminaries

Let be a real quadratic field and be its Dedekind zeta function. Siegel [19] developed a method of computing , where is a positive integer. By specializing Siegel’s formula for a real quadratic field, we obtain the following result.

Theorem 1 (Zagier, [20]). Let be a real quadratic field with discriminant . Then,where denotes the sum of divisors of .

However, there is another method, due to Lang, of computing special values of if is a real quadratic field.

Let be a real quadratic field of discriminant and an ideal class of . Let be any integral ideal belonging to with an integral basis {, }. We putwhere and are the conjugates of and , respectively.

Let be the fundamental unit of . Then, {, } is also integral basis of , and thus we can find a matrix with integer entries satisfying

The entries of are given by

Moreover, det and . See ([13, 14, 21]). Now, we can state Lang’s formula.

Theorem 2 (Lang, [22]). By keeping the abovementioned notation, we havewhere denotes the norm of an ideal , is the norm of , and denotes the generalized Dedekind sum as defined in [23].

Now, we will introduce some basics about quadratic number field . Let be a positive integer and let be a square-free integer. Clearly . In this case, the fundamental unit of is and . We also know that if , where is an odd prime, then splits in as

By ([21], Theorem 2.4), we also know thatwhere will always denote the principal ideal class in .

By ([13], Proposition 3.1), we also know thatwhere C is the ideal class containing or .

3. Main Results

In this section, we will prove our main results. As a start, we record the following theorem that we derive from ([13], Proposition 3.1).

Theorem 3. Let be a positive integer and let be a square-free integer and suppose that(i)(ii) if and if Then,

Now, we find necessary and sufficient condition for to have class number .

Theorem 4. By keeping the abovementioned notation, we have
if and only ifwhere are odd primes, , and .

Proof. Now, by Theorem 1,so thatLet be the ideal class in such that , then by (8), we obtainIf , then 2 remains prime in . We note are distinct nonprincipal ideal classes in
If , then 2 splits in , that is,We assume that denotes the ideal class in such that , then by ([21], Theorem 2.5), we obtainWe note are distinct nonprincipal ideal classes in
Necessary: let . Now, by definition, we haveIf , we obtain , thenThis impliesHence, by (12), we findIf , we obtain , thenThis impliesHence, by (12), we findSufficiency: letThen, by (12), we findBy Theorem 3, we get .
Suppose . Then, there exist at least ideal classes in . Since for any ideal class E, , thusIt is a contradiction.

Data Availability

No date were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Tishreen University.