• Honam Mathematical Journal
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• v.37 no.4
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• pp.469-472
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• 2015

The n-th cyclotomic polynomial ${\Phi}_n(x)$ is irreducible over $\mathbb{Q}$ and has integer coefficients. The degree of ${\Phi}_n(x)$ is ${\varphi}(n)$, where ${\varphi}(n)$ is the Euler Phi-function. In this paper, we define Semi-Cyclotomic Polynomial $J_n(x)$. $J_n(x)$ is also irreducible over $\mathbb{Q}$ and has integer coefficients. But the degree of $J_n(x)$ is $\frac{{\varphi}(n)}{2}$. Galois Theory will be used to prove the above properties of $J_n(x)$.

• Honam Mathematical Journal
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• v.39 no.2
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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.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.3
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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 ${\mathbb{Z}}_n^*/H$.

• Communications of the Korean Mathematical Society
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• v.32 no.1
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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.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.171-177
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• 2019

We associate a positive integer n and a subgroup H of the group G(n) with a polynomial $J_{n,H}(x)$, which is called the Galois polynomial. It turns out that $J_{n,H}(x)$ is a polynomial with integer coefficients for any n and H. In this paper, we provide an equivalent condition for a subgroup H to provide the Galois polynomial which is irreducible over ${\mathbb{Q}}$.

• Honam Mathematical Journal
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• v.40 no.2
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• pp.281-291
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• 2018

We associate a positive integer n and a subgroup H of the group $({\mathbb{Z}}/n{\mathbb{Z}})^{\times}$ with a polynomial $J_n,H(x)$, which is called the Galois polynomial. It turns out that $J_n,H(x)$ is a polynomial with integer coefficients for any n and H. In this paper, we provide an equivalent condition for a subgroup H to provide the Galois polynomial which is irreducible over ${\mathbb{Q}}$ in the case of $n=p^{e_1}_1{\cdots}p^{e_r}_r$ (prime decomposition) with all $e_i{\geq}2$.