A note on prime divisors of polynomials P(T^k), k ≥ 1

2017/05/07 by François Legrand, Legrand, François
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1705.02605

openalex publication_date 2017/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a number field, O_F the integral closure of ℤ in F and P(T) ∈ O_F[T] a monic separable polynomial such that P(0) \not=0 and P(1) \not=0. We give precise sufficient conditions on a given positive integer k for the following condition to hold: there exist infinitely many non-zero prime ideals P of O_F such that the reduction modulo P of P(T) has a root in the residue field O_F/P, but the reduction modulo P of P(T^k) has no root in O_F/P. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers k more precise.

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