Abstract
In this paper, we give an explicit lower bound for the class number of real quadratic field , where is a square-free integer, using which 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 has for a long time inspired the study of lower bounds of .
Many fruitful research studies have been conducted in this direction. Hasse [1] and Yokoi [2, 3] studied lower bounds for class numbers of certain real quadratic fields. Mollin [4, 5] generalized their results for certain real quadratic and biquadratic fields.
In this work, we give a lower bound for , and also we find a necessary and sufficient condition for to have class number .
2. Notation and Preliminaries
Let be a real quadratic field and be its Dedekind zeta function. Siegel [6] 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 [7]). Let be a real quadratic field with discriminant . Thenwhere denotes the sum of divisors of .
However, there is another method, according 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
Now, we can state Lang’s formula.
Theorem 2 (Lang [8]). 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 [9].
To use Lang’s formula, we need to determine the values of , and generalized Dedekind sum.
Lemma 1 (see ref [10]). The entries of are given byMoreover, det and .
Kim [11] obtained the following expressions for generalized Dedekind sum. These expressions are also needed to compute the values of zeta functions for ideal classes of the respective real quadratic fields.
Lemma 2 (Kim [11]). Let be a positive integer. Then, we have(i)(ii)
3. Main Results
Let be a positive integer and let is a square-free integer. Clearly, and is odd. In this case, the fundamental unit of is and . If , then splits in as
By [12], Theorem 2.4, we know thatwhere will always denote the principal ideal class in .
In this section, we will prove our main results. As a start, we record the following proposition.
Proposition 1. Let , where is a square-free integer. Let be an odd prime divisor of , and let C be the ideal class containing or . Then
Proof. Let us assume . Then, is an integral basis for , and thus By Lemma 1, we getSince , Hence, by Lemma 2, we obtainBy Theorem 2, we get
We use Proposition 1 in order to prove the following theorem, which gives a lower bound for .
Theorem 3. Let be a positive integer, and let is a square-free integer. Then
Proof. By [12], Proposition 2.3, it follows that (2) remains prime in
We assume that denotes the principal ideal class in , and then is given by (7). If then splits in as in (6), so let be the ideal class in such that ; then, by Proposition 1, we obtainBy computing , we arrive at . This contradicts the fact that . Similarly, for , and then . This again contradicts the fact that .
We notice are a distinct nonprincipal ideal class in . This completes the proof.
Now, we give a necessary and sufficient condition for to have class number .
Theorem 4. By keeping the abovementioned notation, we have if and only if
Proof. Now, by Theorem 1,If we replace by , where is even, then Now, we replace by , and then we haveNecessary: let . Then, the class group of is . Now, by definition, we haveThis impliesFinally, by (17), we obtainSufficiency: letHence, by (17), we findBy Theorem 3, we get
Suppose . Then, there exist at least ideal classes in .
Since for any ideal class E, ; thus,It is a contradiction.
Data Availability
No data 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.