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On a generalization of Eisenstein's irreducibility criterion

  • Sudesh K. Khanduja (a1) and Jayanti Saha (a1)

Abstract

Let ν be a valuation of any rank of a field K with value group Gν and f(X)= Xm + alXm−1 + … + am be a polynomial over K. In this paper, it is shown that if (ν(ai)/i)≥(ν(am)/m)>0 for l≤im, and there does not exist any integer r>1 dividing m such that ν(am)/rGν, then f(X) is irreducible over K. It is derived as a special case of a more general result proved here. It generalizes the usual Eisenstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields, (cf. [Journal of Number Theory, 52 (1995), 98–118]).

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1.Alexandru, V., Popescu, N. and Zaharescu, A.. A theorem of characterization of residual transcendental extension of a valuation. J. Math. Kyoto Univ., 28 (1988), 579592.
2.Popescu, L. and Popescu, N.. Sur la definition des prolongements residuals transcendents d'une valuation sur un corps K à K(x). Bull. Math. Sci. Math. R. S. Roumanie, 33 (81), No. 3 (1989), 257264.
3.Popescu, N. and Zaharescu, A.. On the structure of irreducible polynomials over local fields. J. Number Theory, 52 (1995), 98118.
4.Zariski, O. and Samuel, P.. Commutative Algebra, Vol. II (D. Van Nostrand Company Inc., Princeton, New Jersey).
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On a generalization of Eisenstein's irreducibility criterion

  • Sudesh K. Khanduja (a1) and Jayanti Saha (a1)

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