• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.351-381
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• 2021

In this paper we introduce the notion of strong Galois H-progenerator object for a finite cocommutative Hopf quasigroup H in a symmetric monoidal category C. We prove that the set of isomorphism classes of strong Galois H-progenerator objects is a subgroup of the group of strong Galois H-objects introduced in [3]. Moreover, we show that strong Galois H-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if H is finite, we find exact sequences of Picard groups related with invertible left H-(quasi)modules and an isomorphism Pic(HMod) ≅ Pic(C)⊕G(H∗) where Pic(HMod) is the Picard group of the category of left H-modules, Pic(C) the Picard group of C, and G(H∗) the group of group-like morphisms of the dual of H.

• Communications of the Korean Mathematical Society
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• v.14 no.1
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• pp.99-110
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• 1999

We apply sector theory to obtain some characterization on Gaois groups for subfactors. As an example of a non-irreducible inclusion of small index, a locally trivial inclusion arising from an automorphism is considered and its Galois group is completely determined by using sector theory.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.657-663
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• 2005

In this paper, we consider groups of permutations S on a set A acting on subsets X of A. In particular, we show that if $X_2{\subseteq}X_1{\subseteq}A$ and Y is an S-normal extension of $X_2 in X_1$, then the Galois group $G_{S}(X_1/Y){\;}of{\;}X_1{\;}over{\;}X_2$ relative to S is an S-closed subgroup of $G_{S}(X_1/X_2)$ in the setting of a Galois theory (correspondence) for this situation.

• East Asian mathematical journal
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• v.38 no.5
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• pp.583-591
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• 2022

Let K be an imaginary quadratic field and 𝒪 be an order in K. Let H_𝒪 be the ring class field of 𝒪. Furthermore, for a positive integer N, let K_𝒪,N be the ray class field modulo N𝒪 of 𝒪. When the discriminant of 𝒪 is different from -3 and -4, we construct an extended form class group which is isomorphic to the Galois group Gal(K_𝒪,N/H_𝒪) and describe its Galois action on K_𝒪,N in a concrete way.

• 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 ${\mathbb{Z}}_n^*/H$.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.199-210
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• 2007

Given a field K and a finite group G, it is a very interesting problem, although very difficult, to find all Galois extensions over K whose Galois group is isomorphic to G. In this paper, we prepare a theoretical background to study this type of problem when G is the Mathieu group $M_{24}$ and K is a field of characteristic two.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.23-31
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• 1996

Our main purpose in this paper is to determine the Galois group of the given irreducible septic polynomial ove Q by using three resolvant polynomials and the discriminant of the given polynomial.

• Bulletin of the Korean Mathematical Society
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• v.44 no.2
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• pp.225-231
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• 2007

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal$({\phi})$. Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope $E[x^{-1}]$ of an inverse polynomial module $M[x^{-1}]$ as a left R[x]-module and we can define an associative Galois group Gal$({\phi}[x^{-1}])$. In this paper we describe the relations between Gal$({\phi})$ and Gal$({\phi}[x^{-1}])$. Then we extend the Galois group of inverse polynomial module and can get Gal$({\phi}[x^{-s}])$, where S is a submonoid of $\mathbb{N}$ (the set of all natural numbers).

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.497-506
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• 2009

This work studies about the composition polynomial f(g(x)) that preserves certain properties of f(x) and g(x). We shall investigate necessary and sufficient conditions of f(x) and g(x) to be f(g(x)) is separable, solvable by radical or split completely. And we find relationship of Galois groups of f(g(x)), f(x) and of g(x).

• Honam Mathematical Journal
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• v.38 no.1
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• pp.85-93
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• 2016

Let A be an abelian variety defined over a number field K and let L be a degree 3 non-Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming that III(A/L) is finite, we compute [III(A/K)][III($A_{\varphi}/K$)]/[III(A/L)], where [X] is the order of a finite abelian group X.