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We show here that for a fixed positive integer $r \\ne 6$ and $n$ sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group $S_{r}$. For $r = 6$, we show the number of exceptional $n \\le N$ for which the Galois group of this polynomial is not $S_r$ is at most $O(\\log N)$."},"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":"1803.02754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-03-07T16:43:30Z","cross_cats_sorted":[],"title_canon_sha256":"bf2d45ec894cb8d0ca9f2e44ec9a5516c6a04af789da2a1d1e8108a4cee18c6a","abstract_canon_sha256":"683f5cb96232b3e3b20f8e6c55807ddf6523750c58354b1604705d4c2d228bb1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:49.308580Z","signature_b64":"EBWbZW2eUwJ54Q8CrESr9Dk/SZfrljy730rbgGLgnEwKhI9dqCtNGZ0qim8nEitDs2oJFOwLYJ0TaLh0GCFaAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b680d00cf22ceb2d3a15f8ca574b0c0f2a8f64b71fadb52ed2d6b0005ab7f0b","last_reissued_at":"2026-05-18T00:21:49.307898Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:49.307898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Galois group over Q of a truncated binomial expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael Filaseta, Richard Moy","submitted_at":"2018-03-07T16:43:30Z","abstract_excerpt":"For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $\\le r$ where $1 \\le r \\le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. We show here that for a fixed positive integer $r \\ne 6$ and $n$ sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group $S_{r}$. 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