• East Asian mathematical journal
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• v.40 no.1
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• pp.95-107
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• 2024

Let K be an imaginary quadratic field with ring of integers 𝓞_K and N be a positive integer. By K_(N) we mean the ray class field of K modulo N𝓞_K. In this paper, for each prime p of K relatively prime to N𝓞_K we explicitly describe the action of the Artin symbol (${\frac{K_{(N)}/K}{p}}$) on special values of modular functions of level N. Furthermore, we extend the Kronecker congruence relation for the elliptic modular function j to some modular functions of higher level.

• Journal of the Korean Mathematical Society
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• v.55 no.2
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• pp.343-372
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• 2018

Let K be an imaginary quadratic field with ring of integers ${\mathcal{O}}_K$. Let E be an elliptic curve with complex multiplication by ${\mathcal{O}}_K$, and let $h_E$ be the Weber function on E. Let $N{\in}\{2,3,4,6\}$. We show that $h_E$ alone when evaluated at a certain N-torsion point on E generates the ray class field of K modulo $N{\mathcal{O}}_K$. This would be a partial answer to the question raised by Hasse and Ramachandra.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.317-326
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• 2021

Let K be an imaginary quadratic field other than ℚ(${\sqrt{-1}}$) and ℚ(${\sqrt{-3}}$), and let 𝒪_K be its ring of integers. Let N be a positive integer such that N = 5 or N ≥ 7. In this paper, we generate the ray class field modulo N𝒪_K over K by using a single x-coordinate of an elliptic curve with complex multiplication by 𝒪_K.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.155-166
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• 2023

In 2018, Andrews introduced the partition functions ${\mathfrak{EO}}$(n) and ${\bar{\mathfrak{EO}}}$(n). The first of these denotes the number of partitions of n in which every even part is less than each odd part, and the second counts the number of partitions enumerated by the first in which only the largest even part appears an odd number of times. In 2021, Pore and Fathima introduced a new partition function ${\mathfrak{EO}}_e$(n) which counts the number of partitions of n which are enumerated by ${\bar{\mathfrak{EO}}}$(n) together with the partitions enumerated by ${\bar{\mathfrak{EO}}}$(n) where all parts are odd and the number of parts is even. They also proved some particular congruences for ${\bar{\mathfrak{EO}}}$(n) and ${\mathfrak{EO}}_e$(n). In this paper, we establish infinitely many families of congruences modulo 2, 4, 5 and 8 for ${\bar{\mathfrak{EO}}}$(n) and modulo 4 for ${\mathfrak{EO}}_e$(n). For example, if p ≥ 5 is a prime with Legendre symbol $({\frac{-3}{p}})=-1$, then for all integers n ≥ 0 and α ≥ 0, we have ${\bar{\mathfrak{EO}}}(8{\cdot}p^{2{\alpha}+1}(pn+j)+{\frac{19{\cdot}p^{2{\alpha}+2}-1}{3}}){\equiv}0$ (mod 8); 1 ≤ j ≤ (p - 1).

• Kyungpook Mathematical Journal
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• v.57 no.4
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• pp.581-600
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• 2017

A classical result of A. Cohn states that, if we express a prime p in base 10 as $$p=a_n10^n+a_{n-1}10^{n-1}+{\cdots}+a_110+a_0$$, then the polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+{\cdots}+a_1x+a_0$ is irreducible in ${\mathbb{Z}}[x]$. This problem was subsequently generalized to any base b by Brillhart, Filaseta, and Odlyzko. We establish this result of A. Cohn in $O_K[x]$, K an imaginary quadratic field such that its ring of integers, $O_K$, is a Euclidean domain. For a Gaussian integer ${\beta}$ with ${\mid}{\beta}{\mid}$ > $1+{\sqrt{2}}/2$, we give another representation for any Gaussian integer using a complete residue system modulo ${\beta}$, and then establish an irreducibility criterion in ${\mathbb{Z}}[i][x]$ by applying this result.

• East Asian mathematical journal
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• v.33 no.3
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• pp.257-263
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• 2017

Let n be any positive integer and ${\mathbb{Z}}_n=\{0,1,{\cdots},n-1\}$ be the ring of integers modulo n. Let $X_n$ be the set of all nonzero, nonunits of ${\mathbb{Z}}_n$, and $G_n$ be the group of all units of ${\mathbb{Z}}_n$. In this paper, by investigating the regular action on $X_n$ by $G_n$, the following are proved : (1) The number of orbits under the regular action (resp. the number of annihilators in $X_n$) is equal to the number of all divisors (${\neq}1$, n) of n; (2) For any positive integer n, ${\sum}_{g{\in}G_n}\;g{\equiv}0$ (mod n); (3) For any orbit o(x) ($x{\in}X_n$) with ${\mid}o(x){\mid}{\geq}2$, ${\sum}_{y{\in}o(x)}\;y{\equiv}0$ (mod n).

• Kyungpook Mathematical Journal
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• v.60 no.4
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• pp.723-729
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• 2020

Let ℤ_n be the ring of integers modulo n and let ℤ_n[X]] be either ℤ_n[X] or ℤ_n[[X]]. Let 𝚪(Z_n[X]]) be the zero-divisor graph of ℤ_n[X]]. In this paper, we study some properties of 𝚪(ℤ_n[X]]). More precisely, we completely characterize the diameter and the girth of 𝚪(ℤ_n[X]]). We also calculate the chromatic number of 𝚪(ℤ_n[X]]).

• East Asian mathematical journal
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• v.37 no.1
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• pp.87-95
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• 2021

Let K be an imaginary quadratic field with ��K its ring of integers. For a positive integer N, let K(N) be the ray class field of K modulo N��K, and let ��N be the field of meromorphic modular functions of level N whose Fourier coefficients lie in the Nth cyclotomic field. For each h ∈ ��N, we construct a ray class invariant as its special value in terms of the extended form class group, and show that the invariant satisfies the natural transformation formula via the Artin map in the sense of Siegel and Stark. Finally, we establish an isomorphism between the extended form class group and Gal(K(N)/K) without any restriction on K.

• Journal of the Korean Mathematical Society
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• v.59 no.3
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• pp.621-634
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• 2022

The scaled inverse of a nonzero element a(x) ∈ ℤ[x]/f(x), where f(x) is an irreducible polynomial over ℤ, is the element b(x) ∈ ℤ[x]/f(x) such that a(x)b(x) = c (mod f(x)) for the smallest possible positive integer scale c. In this paper, we investigate the scaled inverse of (xⁱ - x^j) modulo cyclotomic polynomial of the form Φ_p^s (x) or Φ_p^sq^t (x), where p, q are primes with p < q and s, t are positive integers. Our main results are that the coefficient size of the scaled inverse of (xⁱ - x^j) is bounded by p - 1 with the scale p modulo Φ_p^s (x), and is bounded by q - 1 with the scale not greater than q modulo Φ_p^sq^t (x). Previously, the analogous result on cyclotomic polynomials of the form Φ_2ⁿ (x) gave rise to many lattice-based cryptosystems, especially, zero-knowledge proofs. Our result provides more flexible choice of cyclotomic polynomials in such cryptosystems. Along the way of proving the theorems, we also prove several properties of {x^k}_k∈ℤ in ℤ[x]/Φ_pq(x) which might be of independent interest.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1511-1522
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• 2022

Let H be a subgroup of $\mathbb{Z}^*_n$ (the multiplicative group of integers modulo n) and h₁, h₂, …, h_l distinct representatives of the cosets of H in $\mathbb{Z}^*_n$. We now define a polynomial J_n,H(x) to be $$J_{n,H}(x)=\prod^l_{j=1} \left( x-\sum_{h{\in}H} {\zeta}^{h_jh}_n\right)$$, where ${\zeta}_n=e^{\frac{2{\pi}i}{n}}$ is the nth primitive root of unity. Polynomials of such form generalize the nth cyclotomic polynomial $\Phi_n(x)={\prod}_{k{\in}\mathbb{Z}^*_n}(x-{\zeta}^k_n)$ as J_n,{1}(x) = Φ_n(x). While the nth cyclotomic polynomial Φ_n(x) is irreducible over ℚ, J_n,H(x) is not necessarily irreducible. In this paper, we determine the subgroups H for which J_n,H(x) is irreducible over ℚ.