{"paper":{"title":"A note on prime divisors of polynomials $P(T^k), k \\geq 1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fran\\c{c}ois Legrand","submitted_at":"2017-05-07T12:30:10Z","abstract_excerpt":"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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02605","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}