Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -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 . 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.
Atiyah, M. and Macdonald, I., Introduction to Commutative Algebra, Addison-Wesley Company, 1969.
Atiyah, M. and Macdonald, I., Introduction to Commutative Algebra, Addison-Wesley Company, 1969.)| false