# A note on prime divisors of polynomials $P(T^k), k \geq 1$

## TL;DR

This paper establishes precise conditions on positive integers to identify when prime ideals exist such that a polynomial's reduction has roots but its k-th power polynomial's reduction does not, refining previous results in field arithmetic.

## Contribution

It provides exact criteria for the existence of primes where polynomial reductions behave differently for P(T) and P(T^k), improving earlier general results.

## Key findings

- Identifies conditions on k for prime ideals with specific reduction properties.
- Refines previous results by providing precise sufficient conditions.
- Enhances understanding of prime divisors of polynomial values in number fields.

## Abstract

Let $F$ be a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$ and $P(T) \in O_F[T]$ 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 $\mathcal{P}$ of $O_F$ such that the reduction modulo $\mathcal{P}$ of $P(T)$ has a root in the residue field $O_F/\mathcal{P}$, but the reduction modulo $\mathcal{P}$ of $P(T^k)$ has no root in $O_F/\mathcal{P}$. 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.