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  • 1 Faculty of Sciences Dhar-Mahraz, B.P. 1796, Fes, Morocco
  • | 2 King Khalid University, P.O. Box 9004, Abha, Saudi Arabia
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Abstract

Let R be a discrete valuation ring, p its nonzero prime ideal, PR[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the p-adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo p. By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

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