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

Year, pages: 2019, 213 - 222

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.

Legrand, F. 2019. A note on prime divisors of polynomials $P(T^k), k ≥ 1$. In Mathematica Slovaca, vol. 69, no.1, pp. 213-222. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0215

Legrand, F. (2019). A note on prime divisors of polynomials $P(T^k), k ≥ 1$. Mathematica Slovaca, 69(1), 213-222. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0215