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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1705.02605","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-07T12:30:10Z","cross_cats_sorted":[],"title_canon_sha256":"b8dbe7e1a5874e283a54c67d29e09d4e5698b7e244f055d659f50a3d0462a42a","abstract_canon_sha256":"0018919cc91183ac9bd6a5a72aab9df43650e96ea08d4af07a42392163e23a77"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:18.010174Z","signature_b64":"03p1AZZuTp0u9+KbLC/X69bUTPOpEU8WJDseUwj03iY0ngYW2DKmiy/T6zOMnFRo6RbhsXmOu5lvXchaHY3wDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"799f3d869a634d4c85ff104023590c7b914a3c8fa1f7a0ef24dfe050440c1b14","last_reissued_at":"2026-05-18T00:38:18.009446Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:18.009446Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $O_F/\\mathcal{P}$. 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