• Title/Summary/Keyword: irreducibility criterion

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SOME REMARKS ON EISENSTEIN'S CRITERION

  • Woo, Sung-Sik
    • Communications of the Korean Mathematical Society
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    • v.23 no.4
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    • pp.499-509
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    • 2008
  • In [4] we showed that a polynomial over a Noetherian ring is divisible by some other polynomial by looking at the matrix formed by the coefficients of the polynomials which we called the resultant matrix. Using the result, we will find conditions for a polynomial over a commutative ring to be irreducible. This can be viewed as a generalization of the Eisenstein's irreducibility criterion.

Complete Reducibility of some Modules for a Generalized Kac Moody Lie Algebra

  • Kim, Wansoon
    • Journal of the Chungcheong Mathematical Society
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    • v.5 no.1
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    • pp.195-201
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    • 1992
  • Let G(A) denote a generalized Kac Moody Lie algebra associated to a symmetrizable generalized Cartan matrix A. In this paper, we study on representations of G(A). Highest weight modules and the category O are described. In the main theorem we show that some G(A) modules from the category O are completely reducible. Also a criterion for irreducibility of highest weight modules is obtained. This was proved in [3] for the case of Kac Moody Lie algebras.

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Prime Elements and Irreducible Polynomials over Some Imaginary Quadratic Fields

  • Singthongla, Patiwat;Kanasri, Narakorn Rompurk;Laohakosol, Vichian
    • 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.