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Let $\\alpha$ be a root of \\[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \\] The totally real cyclic fields $K=Q(\\alpha)$ are called simplest sextic fields and are well known in the literature.\n  Using a completely new approach we explicitly give an integral basis of $K$ in a parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$ in which cases we give all generators of power integral bases."},"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":"1809.10072","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-26T15:46:02Z","cross_cats_sorted":[],"title_canon_sha256":"e76bc876ae9323213896aec52aabc89f9c71cee0bca63260ed12396f0b1db6fd","abstract_canon_sha256":"5936d2d79dee12a43b114a49506e0d54b330083103e0ba37327c2d121026013b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:42.665513Z","signature_b64":"OmOYR4TvFU6XWAk3Y4omeKtcUnpfUjVXL8XDx30wvfa+yJ2WUJ6gwuWGUr91kmcPpamtE94Xd1sXtxwnRS4mCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1dfaa276971f9d5ac93c6a5d0cb54a1e80e1abd840d7393e0f47dafa23943724","last_reissued_at":"2026-05-18T00:04:42.664887Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:42.664887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integral bases and monogenity of the simplest sextic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Istv\\'an Ga\\'al, L\\'aszl\\'o Remete","submitted_at":"2018-09-26T15:46:02Z","abstract_excerpt":"Let $m$ be an integer, $m\\neq -8,-3,0,5$ such that $m^2+3m+9$ is square free. Let $\\alpha$ be a root of \\[ f=x^6-2mx^5-(5m+15)x^4-20x^3+5mx^2+(2m+6)x+1. \\] The totally real cyclic fields $K=Q(\\alpha)$ are called simplest sextic fields and are well known in the literature.\n  Using a completely new approach we explicitly give an integral basis of $K$ in a parametric form and we show that the structure of this integral basis is periodic in $m$ with period length 36. We prove that $K$ is not monogenic except for a few values of $m$ in which cases we give all generators of power integral bases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.10072","kind":"arxiv","version":1},"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":"1809.10072","created_at":"2026-05-18T00:04:42.665001+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.10072v1","created_at":"2026-05-18T00:04:42.665001+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.10072","created_at":"2026-05-18T00:04:42.665001+00:00"},{"alias_kind":"pith_short_12","alias_value":"DX5KE5UXD6OV","created_at":"2026-05-18T12:32:19.392346+00:00"},{"alias_kind":"pith_short_16","alias_value":"DX5KE5UXD6OVVSJ4","created_at":"2026-05-18T12:32:19.392346+00:00"},{"alias_kind":"pith_short_8","alias_value":"DX5KE5UX","created_at":"2026-05-18T12:32:19.392346+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/DX5KE5UXD6OVVSJ4NJOQZNKKD2","json":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2.json","graph_json":"https://pith.science/api/pith-number/DX5KE5UXD6OVVSJ4NJOQZNKKD2/graph.json","events_json":"https://pith.science/api/pith-number/DX5KE5UXD6OVVSJ4NJOQZNKKD2/events.json","paper":"https://pith.science/paper/DX5KE5UX"},"agent_actions":{"view_html":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2","download_json":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2.json","view_paper":"https://pith.science/paper/DX5KE5UX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.10072&json=true","fetch_graph":"https://pith.science/api/pith-number/DX5KE5UXD6OVVSJ4NJOQZNKKD2/graph.json","fetch_events":"https://pith.science/api/pith-number/DX5KE5UXD6OVVSJ4NJOQZNKKD2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2/action/storage_attestation","attest_author":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2/action/author_attestation","sign_citation":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2/action/citation_signature","submit_replication":"https://pith.science/pith/DX5KE5UXD6OVVSJ4NJOQZNKKD2/action/replication_record"}},"created_at":"2026-05-18T00:04:42.665001+00:00","updated_at":"2026-05-18T00:04:42.665001+00:00"}