{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:36MTDW5SZ4TAHRPXLWAAOZRAQS","short_pith_number":"pith:36MTDW5S","schema_version":"1.0","canonical_sha256":"df9931dbb2cf2603c5f75d80076620848a461091f80bc957ebd90c14d8e44274","source":{"kind":"arxiv","id":"1608.03935","version":2},"attestation_state":"computed","paper":{"title":"Minkowski's theorem on independent conjugate units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeffrey D. Vaaler, Shabnam Akhtari","submitted_at":"2016-08-13T03:45:51Z","abstract_excerpt":"We call a unit $\\beta$ in a Galois extension $l/\\mathbb{Q}$ a Minkowski unit if the subgroup generated by $\\beta$ and its conjugates over $\\mathbb{Q}$ has maximum rank in the unit group of $l$. Minkowski showed the existence of such units in every Galois extension. We will give a new proof to Minkowski's theorem and show that there exists a Minkowski unit $\\beta \\in l$ such that the Weil height of $\\beta$ is comparable with the sum of the heights of a fundamental system of units of $l$. Our proof implies a bound on the index of the subgroup generated by the algebraic conjugates of $\\beta$ in t"},"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":"1608.03935","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-13T03:45:51Z","cross_cats_sorted":[],"title_canon_sha256":"6390006cafadf6e0c2ebfb2eb9574b85e8b54c5efcee5ae4c24d369f26d36b06","abstract_canon_sha256":"f7fca4ea24c8317a24455b769c52137c6fe1d05541b98802f53be5b5fcaa8d9c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:22.898286Z","signature_b64":"ERkkfnAuqc3im63RMBJn86U9lm70ndPMz08Qb6fK/kbDfGv8RUmyhCzShXwD7NCjsPip5fE87byAdismNyRhCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df9931dbb2cf2603c5f75d80076620848a461091f80bc957ebd90c14d8e44274","last_reissued_at":"2026-05-18T00:52:22.897620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:22.897620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minkowski's theorem on independent conjugate units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeffrey D. Vaaler, Shabnam Akhtari","submitted_at":"2016-08-13T03:45:51Z","abstract_excerpt":"We call a unit $\\beta$ in a Galois extension $l/\\mathbb{Q}$ a Minkowski unit if the subgroup generated by $\\beta$ and its conjugates over $\\mathbb{Q}$ has maximum rank in the unit group of $l$. Minkowski showed the existence of such units in every Galois extension. We will give a new proof to Minkowski's theorem and show that there exists a Minkowski unit $\\beta \\in l$ such that the Weil height of $\\beta$ is comparable with the sum of the heights of a fundamental system of units of $l$. Our proof implies a bound on the index of the subgroup generated by the algebraic conjugates of $\\beta$ in t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03935","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":"1608.03935","created_at":"2026-05-18T00:52:22.897763+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.03935v2","created_at":"2026-05-18T00:52:22.897763+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.03935","created_at":"2026-05-18T00:52:22.897763+00:00"},{"alias_kind":"pith_short_12","alias_value":"36MTDW5SZ4TA","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"36MTDW5SZ4TAHRPX","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"36MTDW5S","created_at":"2026-05-18T12:29:55.572404+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/36MTDW5SZ4TAHRPXLWAAOZRAQS","json":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS.json","graph_json":"https://pith.science/api/pith-number/36MTDW5SZ4TAHRPXLWAAOZRAQS/graph.json","events_json":"https://pith.science/api/pith-number/36MTDW5SZ4TAHRPXLWAAOZRAQS/events.json","paper":"https://pith.science/paper/36MTDW5S"},"agent_actions":{"view_html":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS","download_json":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS.json","view_paper":"https://pith.science/paper/36MTDW5S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.03935&json=true","fetch_graph":"https://pith.science/api/pith-number/36MTDW5SZ4TAHRPXLWAAOZRAQS/graph.json","fetch_events":"https://pith.science/api/pith-number/36MTDW5SZ4TAHRPXLWAAOZRAQS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS/action/storage_attestation","attest_author":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS/action/author_attestation","sign_citation":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS/action/citation_signature","submit_replication":"https://pith.science/pith/36MTDW5SZ4TAHRPXLWAAOZRAQS/action/replication_record"}},"created_at":"2026-05-18T00:52:22.897763+00:00","updated_at":"2026-05-18T00:52:22.897763+00:00"}