• 대한수학회지
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• 제35권2호
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• pp.341-360
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• 1998

Some interesting properties of Carlitz cyclotomic polynomials analogous to those of classical cyclotomic polynomials are given.

• 충청수학회지
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• 제31권3호
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• pp.309-319
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• 2018

Galois polynomials are defined as a generalization of the cyclotomic polynomials. The definition of Galois polynomials (and cyclotomic polynomials) is based on the multiplicative group of integers modulo n, i.e. ${\mathbb{Z}}_n^*$. In this paper, we define Galois polynomials which are based on the quotient group ${\mathbb{Z}}_n^*/H$.

• 호남수학학술지
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• 제39권2호
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• pp.259-265
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• 2017

Galois polynomials are defined as a generalization of the Cyclotomic polynomials. Galois polynomials have integer coefficients as the cyclotomic polynomials. But they are not always irreducible. In this paper, Galois polynomials are partly classified according to the type of subgroups which defines the Galois polynomial.

• 대한수학회논문집
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• 제27권3호
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• pp.449-454
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• 2012

In this paper, we find the minimal polynomial of a primitive root of unity over cyclotomic fields. From this, we factorize cyclotomic polynomials over cyclotomic fields and investigate the coefficients of ${\Phi}_{3n}(x)$ when 3∤$n$.

• 대한수학회보
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• 제54권1호
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• pp.43-50
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• 2017

Let A(n) denote the largest absolute value of the coefficients of n-th cyclotomic polynomial ${\Phi}_n(x)$ and let p < q < r be odd primes. In this note, we give an infinite family of cyclotomic polynomials ${\Phi}_{pqr}(x)$ with A(pqr) = 3, without fixing p.

• 대한수학회논문집
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• 제28권1호
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• pp.25-38
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• 2013

In this paper we compute the resultants of the Carlitz cyclotomic polynomials and then we address two applications to the setting of the Carlitz module.

• 대한수학회논문집
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• 제32권1호
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• pp.1-6
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• 2017

In this paper, the fundamental theorem of Galois Theory is used to generalize cyclotomic polynomials and construct irreducible polynomials associated with the n-th primitive roots of unity.

• 대한수학회보
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• 제52권6호
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• pp.1911-1924
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• 2015

A cyclotomic polynomial ${\Phi}_n(x)$ is said to be ternary if n = pqr for three distinct odd primes p < q < r. Let A(n) be the largest absolute value of the coefficients of ${\Phi}_n(x)$. If A(n) = 1 we say that ${\Phi}_n(x)$ is flat. In this paper, we classify all flat ternary cyclotomic polynomials ${\Phi}_{pqr}(x)$ in the case $q{\equiv}{\pm}1$ (mod p) and $4r{\equiv}{\pm}1$ (mod pq).

• 대한수학회보
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• 제59권6호
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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.

• 대한수학회보
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• 제51권4호
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• pp.949-955
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• 2014

Let ${\Phi}_n(x)={\sum}^{{\phi}(n)}_{k=0}a(n,k)x^k$ denote the n-th cyclotomic polynomial. In this note, let p < q < r be odd primes, where $q{\not{\equiv}}1$ (mod p) and $r{\equiv}-2$ (mod pq), we construct an explicit k such that a(pqr, k) = -2.