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In this note, we prove that outside of some natural exceptional set $T\\subset \\O_{k,S}^*$, the prime ideals of $\\O_k$ dividing $f(u)$, $u\\in \\O_{k,S}^*\\setminus T$, mostly have degree one over $\\Q$; that is, the corresponding residue fields have degree one over the prime field. We also formulate a conjectural analogue of this result for rational points on an elliptic curve over a number field, and deduce our "},"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":"1106.1385","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-06-07T16:07:39Z","cross_cats_sorted":[],"title_canon_sha256":"c378368fbc27b6ad4c8e0ee5a6f54d79693a6d6b06056effdab493358d74989f","abstract_canon_sha256":"f9baf2b9f708437a1c4eb58166c7439d69d933d10f9de375ac214f67ae49d948"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:23.799775Z","signature_b64":"t2z2JUnmqubE88jgYWih2kvoNrGAfkifVJXovlZfv1awkPLHWNmRk0Lab2TzCOR1Os2buE61Rrf30/0iuR/4Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e39f5221508a7ba1dcc7b81e994c3593903018cd5a1efea0808b248c3222e44","last_reissued_at":"2026-05-18T04:20:23.798988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:23.798988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ideals of degree one contribute most of the height","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aaron Levin, David McKinnon","submitted_at":"2011-06-07T16:07:39Z","abstract_excerpt":"Let $k$ be a number field, $f(x)\\in k[x]$ a polynomial over $k$ with $f(0)\\neq 0$, and $\\O_{k,S}^*$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural exceptional set $T\\subset \\O_{k,S}^*$, the prime ideals of $\\O_k$ dividing $f(u)$, $u\\in \\O_{k,S}^*\\setminus T$, mostly have degree one over $\\Q$; that is, the corresponding residue fields have degree one over the prime field. 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