• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.559-578
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• 2019

We list all subfields of cyclotomic function fields over rational function fields with class number three. We also determine all the imaginary abelian extensions with relative class number three, explicitly.

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.83-89
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• 1998

Using the list of all imaginary quadratic fields with class number 1, 2, 3 and 6, we determine all imaginary bicyclic biquadratic number fields of class number 3. There are exactly 163 such fields and their conductors are less than or equal to 163 $\cdot$883.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1559-1568
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• 2015

The relative class number $H_d(f)$ of a real quadratic field $K=\mathbb{Q}(\sqrt{m})$ of discriminant d is the ratio of class numbers of $O_f$ and $O_K$, where $O_K$ denotes the ring of integers of K and $O_f$ is the order of conductor f given by $\mathbb{Z}+fO_K$. In a recent paper of A. Furness and E. A. Parker the relative class number of $\mathbb{Q}(\sqrt{m})$ has been investigated using continued fraction in the special case when $(\sqrt{m})$ has a diagonal form. Here, we extend their result and show that there exists a conductor f of relative class number 1 when the continued fraction of $(\sqrt{m})$ is non-diagonal of period 4 or 5. We also show that there exist infinitely many real quadratic fields with any power of 2 as relative class number if there are infinitely many Mersenne primes.

• Journal of Technologic Dentistry
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• v.29 no.2
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• pp.197-211
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• 2007

With the development of orthodontics and increasing concerns on physical appearance, the number of patients has been steadily increasing. It is quite important not only to make effective cure plans and accurate diagnoses but also to have a thorough grasp of patients' malocclusion types and their occurrence frequency, in addition to patients' personality in order to cure the patients appropriately. This study is based on 946 malocclusion patients who had visited Chong-A Dental Hospital from 1999 to 2004 and investigated their aspects of malocclusion and characteristics of their gender, age and residence. The results are as follows. 1. The number of patients per year had been decreased until 2001, after which year the number had fluctuated. The number was the largest in 1999, 169 and the smallest in 2001, 140. Female occupied 68.0% of the total, twice as many as male, 32.0%) 2. Based on the Angle's classification, 19 or over year - old group was the largest of the total, 59.3% and 6 or younger year - old group, the smallest, 0.5%. The 19 or over year old group was less than a half of the total (47.4%) in 2003 and there were no patients who belonged to the 6 or younger year - old group in 2003 and 2004. 3. Distributions on the types of malocclusion have shown that 39.9 % of the total are in the Class I, the largest, 31.0% in the Class I and 29.2 in the Class II, the smallest. 1) The number of the ClassI was 73, the largest, that of the Class III being 35, the smallest in 1999. On the whole, the number of the Class I accounted for the largest part of the total. 2) The number of male patients in the Class II was the smallest, generally being the largest in the Class I. In case of female, that of the Class III was the smallest. 3) Based on the age, the Class I was the highest in between 7 and 13 age group, the Class III the lowest. The Class I occupied the largest around 40%. 4) In the shape of physiognomy, the meso occupied the largest part among all the Class, of which the Class II was the highest, 64.2%. The bracy was the largest in the Class I, and the dolicho in the Class III. 5) In the profile, the convex shape was the largest in the Class I and II, and especially in the Class II, over 3/4 of the total, 75.4%. In contrast, the direct shape was the largest in the Class III and the sunken shape occupied 33.3%, which was nearly ten times more than the case of the Class I and III. 6) In the asymmetry of physiognomy, the number of patients of the Class IIIwas the largest, 34.1% and that of the Class II, the smallest, 19.5%. It was found that about one fourth of the malocclusion patients were under the asymmetry of physiognomy. 4. In the distribution of patients' residence, 81.4% were from the Seoul Metropolis and 48.2% from Gangnam-Gu where Chong-A Dental Hospital is located and Seocho-Gu and Songpa-Gu which are adjacent to Gangnam-Gu.

• East Asian mathematical journal
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• v.37 no.5
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• pp.613-617
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• 2021

Abstract. For a positive square-free integer m, let K = ℚ($\sqrt{m}$) be a real quadratic field. The relative class number H_d(f) of K of discriminant d is the ratio of class numbers 𝒪_K and 𝒪_f, where 𝒪_K is the ring of integers of K and 𝒪_f is the order of conductor f given by ℤ + f𝒪_K. In 1856, Dirichlet showed that for certain m there exists an infinite number of f such that the relative class number H_d(f) is one. But it remained open as to whether there exists such an f for each m. In this paper, we give a result for existence of real quadratic field ℚ($\sqrt{m}$) with relative class number one where the period of continued fraction expansion of $\sqrt{m}$ is 6.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.799-814
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• 2009

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, using Shimura Reciprocity Law, we compute the Galois actions of some class invariants from the generalized Weber functions $\mathfrak{g}_0,\mathfrak{g}_1,\mathfrak{g}_2$ and $\mathfrak{g}_3$ over quadratic number fields with discriminant $D{\equiv}1$ (mod 12).

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.315-323
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• 2023

First, we recall the results for prime-producing polynomials related to class number one problem of quadratic fields. Next, we give the relation between prime-producing cubic polynomials and class number one problem of the simplest cubic fields and then present the conjecture for the relations. Finally, we numerically compare the ratios producing prime values for several polynomials in some interval.

• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.595-606
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• 2001

Using p-adic class number formula, we derive a congru-ence relation for class numbers of the simplest cubic fields which can be considered as a cubic analogue of Ankeny-Artin-Chowlas theo-rem, Furthermore, we give an elementary proof for an upper bound for the class numbers of the simplest cubic fields.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.1
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• pp.213-219
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• 2013

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, we compute the Galois actions of a class invariant from a generalized Weber function $g_1$ over imaginary quadratic number fields with discriminant $D{\equiv}64(mod72)$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.921-925
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• 2011

A class invariant is the value of a modular function that generates a ring class field of an imaginary quadratic number field such as the singular moduli of level 1. In this paper, using Shimura Reciprocity Law, we compute the Galois actions of a class invariant from a generalized Weber function $g_2$ over quadratic number fields with discriminant $D{\equiv}21$ (mod 36).