2018/10/06 by M. E. Charkani, Charkani, M. E., Abdulaziz Deajim +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1810.02985
Let R be a Dedekind ring, \mathfrakp a nonzero prime ideal of R, P∈ R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the \mathfrakp-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 \mathfrakp. As an important application, when the lower bound is greater than zero for some \mathfrakp, we conclude that no root of P generates a power integral basis in the field extension of K defined by P.