{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:NY6UFKCTVGDHSZV5FLVFPYPZ4K","short_pith_number":"pith:NY6UFKCT","canonical_record":{"source":{"id":"1611.04103","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-13T08:44:40Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"8536e4f4ad19725027e99e8c86cd8288a4b52ffee1c8418d1b6722f0f846370b","abstract_canon_sha256":"5572239e1279777e0812837dc4566cd3930b76f2f1dba861705006714348fef7"},"schema_version":"1.0"},"canonical_sha256":"6e3d42a853a9867966bd2aea57e1f9e2b9d232e9b3447813b2bece87749206c7","source":{"kind":"arxiv","id":"1611.04103","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04103","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04103v1","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04103","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"pith_short_12","alias_value":"NY6UFKCTVGDH","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"NY6UFKCTVGDHSZV5","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"NY6UFKCT","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:NY6UFKCTVGDHSZV5FLVFPYPZ4K","target":"record","payload":{"canonical_record":{"source":{"id":"1611.04103","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-13T08:44:40Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"8536e4f4ad19725027e99e8c86cd8288a4b52ffee1c8418d1b6722f0f846370b","abstract_canon_sha256":"5572239e1279777e0812837dc4566cd3930b76f2f1dba861705006714348fef7"},"schema_version":"1.0"},"canonical_sha256":"6e3d42a853a9867966bd2aea57e1f9e2b9d232e9b3447813b2bece87749206c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:17.552687Z","signature_b64":"5jJhhc3t1q5JMZ2uiwDaTcub6gqjysVq8EBt1+1xvb483OCDUse7oUc2YEr0vr9cACF12QFw8SETCyBzyzeICg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e3d42a853a9867966bd2aea57e1f9e2b9d232e9b3447813b2bece87749206c7","last_reissued_at":"2026-05-18T00:59:17.551969Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:17.551969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.04103","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:59:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jjz66JiGA5oreufHtpGTLaQoDpDEFrCFXq2xYvo4/Xh6ESyV2+ZwkRwPHfI0nXNmvqpW19uuWAj7yDGNsNgFAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:24:57.533883Z"},"content_sha256":"1651f12a6bbc29b1d1fbfb3f79a5dc219a5c309192e2fd106017434183ce08ae","schema_version":"1.0","event_id":"sha256:1651f12a6bbc29b1d1fbfb3f79a5dc219a5c309192e2fd106017434183ce08ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:NY6UFKCTVGDHSZV5FLVFPYPZ4K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Upper bound on the number of ramified primes for odd order solvable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Daniel Rabayev","submitted_at":"2016-11-13T08:44:40Z","abstract_excerpt":"Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\\mathbb{Q}$ with exactly $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04103","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:59:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VaW1pumNkvEsPRcXqiNVfbhVLhn83yo5kAKqgOVADuGJYjq5L3dPTHTmqO3I7KQ8tCFDvmLR/ZnFD/xV86LFCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:24:57.534541Z"},"content_sha256":"e57c839610c18fcf928613be8351521b98e27c864613815b6f6db60c644e011a","schema_version":"1.0","event_id":"sha256:e57c839610c18fcf928613be8351521b98e27c864613815b6f6db60c644e011a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NY6UFKCTVGDHSZV5FLVFPYPZ4K/bundle.json","state_url":"https://pith.science/pith/NY6UFKCTVGDHSZV5FLVFPYPZ4K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NY6UFKCTVGDHSZV5FLVFPYPZ4K/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T09:24:57Z","links":{"resolver":"https://pith.science/pith/NY6UFKCTVGDHSZV5FLVFPYPZ4K","bundle":"https://pith.science/pith/NY6UFKCTVGDHSZV5FLVFPYPZ4K/bundle.json","state":"https://pith.science/pith/NY6UFKCTVGDHSZV5FLVFPYPZ4K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NY6UFKCTVGDHSZV5FLVFPYPZ4K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:NY6UFKCTVGDHSZV5FLVFPYPZ4K","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5572239e1279777e0812837dc4566cd3930b76f2f1dba861705006714348fef7","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-13T08:44:40Z","title_canon_sha256":"8536e4f4ad19725027e99e8c86cd8288a4b52ffee1c8418d1b6722f0f846370b"},"schema_version":"1.0","source":{"id":"1611.04103","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04103","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04103v1","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04103","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"pith_short_12","alias_value":"NY6UFKCTVGDH","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"NY6UFKCTVGDHSZV5","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"NY6UFKCT","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:e57c839610c18fcf928613be8351521b98e27c864613815b6f6db60c644e011a","target":"graph","created_at":"2026-05-18T00:59:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\\mathbb{Q}$ with exactly $","authors_text":"Daniel Rabayev","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-13T08:44:40Z","title":"Upper bound on the number of ramified primes for odd order solvable groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04103","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1651f12a6bbc29b1d1fbfb3f79a5dc219a5c309192e2fd106017434183ce08ae","target":"record","created_at":"2026-05-18T00:59:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5572239e1279777e0812837dc4566cd3930b76f2f1dba861705006714348fef7","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-13T08:44:40Z","title_canon_sha256":"8536e4f4ad19725027e99e8c86cd8288a4b52ffee1c8418d1b6722f0f846370b"},"schema_version":"1.0","source":{"id":"1611.04103","kind":"arxiv","version":1}},"canonical_sha256":"6e3d42a853a9867966bd2aea57e1f9e2b9d232e9b3447813b2bece87749206c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6e3d42a853a9867966bd2aea57e1f9e2b9d232e9b3447813b2bece87749206c7","first_computed_at":"2026-05-18T00:59:17.551969Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:59:17.551969Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5jJhhc3t1q5JMZ2uiwDaTcub6gqjysVq8EBt1+1xvb483OCDUse7oUc2YEr0vr9cACF12QFw8SETCyBzyzeICg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:59:17.552687Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.04103","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1651f12a6bbc29b1d1fbfb3f79a5dc219a5c309192e2fd106017434183ce08ae","sha256:e57c839610c18fcf928613be8351521b98e27c864613815b6f6db60c644e011a"],"state_sha256":"a6398d78163f6ccf0abcf497d55928ef5467f61edc6a24abd261253bed38968b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NR3PSJZOYjSRoyE5If6TAGLu0AMGX3iaByTGYP+fndaSJA22FtADR+3xLq3S6jQNyRpLrt0787MpAQFHFgSfAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:24:57.538150Z","bundle_sha256":"edcda87aca811c5075127834861fc75158249a369ac629f38a4f4475dd9f102a"}}