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For example, suppose $K$ is a number field and $f\\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $\\beta\\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$."},"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":"1605.04376","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-05-14T04:22:06Z","cross_cats_sorted":[],"title_canon_sha256":"b7611a376615f94a653feb134aca65d7a5513d4b51cd23e043154954c5dd23a9","abstract_canon_sha256":"212523ee6e62fa955cdadf28ab14d250baec1e93ab5749166eb2537fccac2671"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:09.855709Z","signature_b64":"INZgR6MGtYEVQjGHXhwIv7yIa2pMaH9Sx+TtxF0dJtseKxBpFtS1HYTXquuWD27rpO9szSXjO8UXttf4sojQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1988ed47fed971bfab5daf3dc16fedaf92434d9cb3b7d59920c1589eb468e471","last_reissued_at":"2026-05-18T00:48:09.855012Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:09.855012Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"ABC implies a Zsigmondy principle for ramification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew Bridy, Thomas Tucker","submitted_at":"2016-05-14T04:22:06Z","abstract_excerpt":"Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in $K(x)$ that are not postcritically finite. For example, suppose $K$ is a number field and $f\\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $\\beta\\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04376","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1605.04376","created_at":"2026-05-18T00:48:09.855111+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.04376v2","created_at":"2026-05-18T00:48:09.855111+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04376","created_at":"2026-05-18T00:48:09.855111+00:00"},{"alias_kind":"pith_short_12","alias_value":"DGEO2R763FY3","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"DGEO2R763FY37K25","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"DGEO2R76","created_at":"2026-05-18T12:30:12.583610+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6","json":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6.json","graph_json":"https://pith.science/api/pith-number/DGEO2R763FY37K25V464C37NV6/graph.json","events_json":"https://pith.science/api/pith-number/DGEO2R763FY37K25V464C37NV6/events.json","paper":"https://pith.science/paper/DGEO2R76"},"agent_actions":{"view_html":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6","download_json":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6.json","view_paper":"https://pith.science/paper/DGEO2R76","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.04376&json=true","fetch_graph":"https://pith.science/api/pith-number/DGEO2R763FY37K25V464C37NV6/graph.json","fetch_events":"https://pith.science/api/pith-number/DGEO2R763FY37K25V464C37NV6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6/action/storage_attestation","attest_author":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6/action/author_attestation","sign_citation":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6/action/citation_signature","submit_replication":"https://pith.science/pith/DGEO2R763FY37K25V464C37NV6/action/replication_record"}},"created_at":"2026-05-18T00:48:09.855111+00:00","updated_at":"2026-05-18T00:48:09.855111+00:00"}