• Title, Summary, Keyword: Galois groups

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ON GALOIS GROUPS FOR NON-IRREDUCIBLE INCLUSIONS OF SUBFACTORS

  • Lee, Jung-Rye
    • 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.

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On the galois groups of the septic polynomials

  • Lee, Geon-No
    • 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.

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GALOIS POLYNOMIALS FROM QUOTIENT GROUPS

  • Lee, Ki-Suk;Lee, Ji-eun;Brandli, Gerold;Beyne, Tim
    • 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 ${\mathbb{Z}}_n^*/H$.

GALOIS THEORY OF MATHIEU GROUPS IN CHARACTERISTIC TWO

  • Yie, Ik-Kwon
    • 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.

Design of Non-Binary Quasi-Cyclic LDPC Codes Based on Multiplicative Groups and Euclidean Geometries

  • Jiang, Xueqin;Lee, Moon-Ho
    • Journal of Communications and Networks
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    • v.12 no.5
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    • pp.406-410
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    • 2010
  • This paper presents an approach to the construction of non-binary quasi-cyclic (QC) low-density parity-check (LDPC) codes based on multiplicative groups over one Galois field GF(q) and Euclidean geometries over another Galois field GF($2^S$). Codes of this class are shown to be regular with girth $6{\leq}g{\leq}18$ and have low densities. Finally, simulation results show that the proposed codes perform very wel with the iterative decoding.

GALOIS GROUPS OF MODULES AND INVERSE POLYNOMIAL MODULES

  • Park, Sang-Won;Jeong, Jin-Sun
    • 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).

COMPOSITION OF POLYNOMIALS OVER A FIELD

  • Choi, EunMi
    • 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).

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GALOIS STRUCTURES OF DEFINING FIELDS OF FAMILIES OF ELLIPTIC CURVES WITH CYCLIC TORSION

  • Jeon, Daeyeol
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.2
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    • pp.205-210
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    • 2014
  • The author with C. H. Kim and Y. Lee constructed infinite families of elliptic curves over cubic number fields K with prescribed torsion groups which occur infinitely often. In this paper, we examine the Galois structures of such cubic number fields K for the families of elliptic curves with cyclic torsion.

ON THE TATE-SHAFAREVICH GROUPS OVER DEGREE 3 NON-GALOIS EXTENSIONS

  • Yu, Hoseog
    • 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.