{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:QRVEHNWA7U2LW4NZDYXCAIPS5Q","short_pith_number":"pith:QRVEHNWA","canonical_record":{"source":{"id":"1802.09599","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-26T20:53:04Z","cross_cats_sorted":[],"title_canon_sha256":"d72816d977e994b5e933fe5226a124b3064dfe3c9d07c78c542a80c8fb85bbd7","abstract_canon_sha256":"890fdeb514034ff3c72379dd9537037f49670c9770706b7431f901c902b91430"},"schema_version":"1.0"},"canonical_sha256":"846a43b6c0fd34bb71b91e2e2021f2ec069d7bd5fcae0ce02957c33e68aa4dd4","source":{"kind":"arxiv","id":"1802.09599","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.09599","created_at":"2026-05-18T00:06:40Z"},{"alias_kind":"arxiv_version","alias_value":"1802.09599v2","created_at":"2026-05-18T00:06:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.09599","created_at":"2026-05-18T00:06:40Z"},{"alias_kind":"pith_short_12","alias_value":"QRVEHNWA7U2L","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QRVEHNWA7U2LW4NZ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QRVEHNWA","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:QRVEHNWA7U2LW4NZDYXCAIPS5Q","target":"record","payload":{"canonical_record":{"source":{"id":"1802.09599","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-26T20:53:04Z","cross_cats_sorted":[],"title_canon_sha256":"d72816d977e994b5e933fe5226a124b3064dfe3c9d07c78c542a80c8fb85bbd7","abstract_canon_sha256":"890fdeb514034ff3c72379dd9537037f49670c9770706b7431f901c902b91430"},"schema_version":"1.0"},"canonical_sha256":"846a43b6c0fd34bb71b91e2e2021f2ec069d7bd5fcae0ce02957c33e68aa4dd4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:40.602801Z","signature_b64":"qd9kwGvr2mATS1f5KalAWVPLTHJ7kL/Q92W4YJhqsfRI141cuQd0UtOEr/Q4lNk8UZIPUus5vEl1EPYzZzTMAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"846a43b6c0fd34bb71b91e2e2021f2ec069d7bd5fcae0ce02957c33e68aa4dd4","last_reissued_at":"2026-05-18T00:06:40.602425Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:40.602425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.09599","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j/zBnWfTOcl4l5FVkDDXxlvc1MpRwLEDtv9q+PUauzRwd6hJhw+KPTBOzUd3eISGRqj/vnYvrTrTyjv08gfJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T04:01:07.398176Z"},"content_sha256":"aa93b8cf65804c109f14348a0c5124fd351cf56d86719947b89ba3bb1260ff95","schema_version":"1.0","event_id":"sha256:aa93b8cf65804c109f14348a0c5124fd351cf56d86719947b89ba3bb1260ff95"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:QRVEHNWA7U2LW4NZDYXCAIPS5Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two Families of Monogenic $S_4$ Quartic Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hanson Smith","submitted_at":"2018-02-26T20:53:04Z","abstract_excerpt":"Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\\dfrac{256b^3-27a^4}{\\gcd(256b^3,27a^4)}$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a monogenic extension of $\\mathbb{Q}$ and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generatin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09599","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:06:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hX6zbRUEoaDKE0MRCgKdz7uXvnsCwcbXncJRH1St1zcw8yMrmwETMTxbbTpiM21QRlWKdfZLdgEnRZyqK2ZMBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T04:01:07.398534Z"},"content_sha256":"5460a155164c0f190ed5c223baa54d0eb3a865a92a407f1ff28b2f0447128988","schema_version":"1.0","event_id":"sha256:5460a155164c0f190ed5c223baa54d0eb3a865a92a407f1ff28b2f0447128988"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QRVEHNWA7U2LW4NZDYXCAIPS5Q/bundle.json","state_url":"https://pith.science/pith/QRVEHNWA7U2LW4NZDYXCAIPS5Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QRVEHNWA7U2LW4NZDYXCAIPS5Q/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":"2026-06-23T04:01:07Z","links":{"resolver":"https://pith.science/pith/QRVEHNWA7U2LW4NZDYXCAIPS5Q","bundle":"https://pith.science/pith/QRVEHNWA7U2LW4NZDYXCAIPS5Q/bundle.json","state":"https://pith.science/pith/QRVEHNWA7U2LW4NZDYXCAIPS5Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QRVEHNWA7U2LW4NZDYXCAIPS5Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QRVEHNWA7U2LW4NZDYXCAIPS5Q","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":"890fdeb514034ff3c72379dd9537037f49670c9770706b7431f901c902b91430","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-26T20:53:04Z","title_canon_sha256":"d72816d977e994b5e933fe5226a124b3064dfe3c9d07c78c542a80c8fb85bbd7"},"schema_version":"1.0","source":{"id":"1802.09599","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.09599","created_at":"2026-05-18T00:06:40Z"},{"alias_kind":"arxiv_version","alias_value":"1802.09599v2","created_at":"2026-05-18T00:06:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.09599","created_at":"2026-05-18T00:06:40Z"},{"alias_kind":"pith_short_12","alias_value":"QRVEHNWA7U2L","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QRVEHNWA7U2LW4NZ","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QRVEHNWA","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:5460a155164c0f190ed5c223baa54d0eb3a865a92a407f1ff28b2f0447128988","target":"graph","created_at":"2026-05-18T00:06:40Z","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":"Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\\dfrac{256b^3-27a^4}{\\gcd(256b^3,27a^4)}$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a monogenic extension of $\\mathbb{Q}$ and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generatin","authors_text":"Hanson Smith","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-26T20:53:04Z","title":"Two Families of Monogenic $S_4$ Quartic Number Fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09599","kind":"arxiv","version":2},"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:aa93b8cf65804c109f14348a0c5124fd351cf56d86719947b89ba3bb1260ff95","target":"record","created_at":"2026-05-18T00:06:40Z","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":"890fdeb514034ff3c72379dd9537037f49670c9770706b7431f901c902b91430","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-26T20:53:04Z","title_canon_sha256":"d72816d977e994b5e933fe5226a124b3064dfe3c9d07c78c542a80c8fb85bbd7"},"schema_version":"1.0","source":{"id":"1802.09599","kind":"arxiv","version":2}},"canonical_sha256":"846a43b6c0fd34bb71b91e2e2021f2ec069d7bd5fcae0ce02957c33e68aa4dd4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"846a43b6c0fd34bb71b91e2e2021f2ec069d7bd5fcae0ce02957c33e68aa4dd4","first_computed_at":"2026-05-18T00:06:40.602425Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:40.602425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qd9kwGvr2mATS1f5KalAWVPLTHJ7kL/Q92W4YJhqsfRI141cuQd0UtOEr/Q4lNk8UZIPUus5vEl1EPYzZzTMAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:40.602801Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.09599","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aa93b8cf65804c109f14348a0c5124fd351cf56d86719947b89ba3bb1260ff95","sha256:5460a155164c0f190ed5c223baa54d0eb3a865a92a407f1ff28b2f0447128988"],"state_sha256":"04b15a429967d1a516c028eb7b9c3746d95632002d8407e12c16d7235be178e6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3QZbDlIvsv0gDRQInTGLXAKTcMXauaMja+VkSIn+ejhOluw/BcriJjcPBIq7sSq4HHfM2fkjnM9+HminPxb+BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T04:01:07.400387Z","bundle_sha256":"2a39c45bc3a28eb65e1f81f171276c170e57d658b7c134e53f939cc6d9aecd85"}}