{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QQRMMKZ3CRTH24W25L6N4SSOI5","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":"6a9de35ec61b4f1d37c6608b7c170b4e9da13428dee1f0298a39ad73555c2d14","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-27T03:04:30Z","title_canon_sha256":"09b8217c566b74912ac806173ee4bf5b7718f8087f6504f1d9962308188ce681"},"schema_version":"1.0","source":{"id":"1806.11052","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.11052","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"arxiv_version","alias_value":"1806.11052v1","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.11052","created_at":"2026-05-18T00:12:07Z"},{"alias_kind":"pith_short_12","alias_value":"QQRMMKZ3CRTH","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QQRMMKZ3CRTH24W2","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QQRMMKZ3","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:f35ba491b7365000e7461328447ec3874a0c9006b55d86dd6797309a5f5774ae","target":"graph","created_at":"2026-05-18T00:12:07Z","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 $\\mathcal{S}_q$ denote the group of all square elements in the multiplicative group $\\mathbb{F}_q^*$ of a finite field $\\mathbb{F}_q$ of odd characteristic containing $q$ elements. Let $\\mathcal{O}_q$ be the set of all odd order elements of $\\mathbb{F}_q^*$. Then $\\mathcal{O}_q$ turns up as a subgroup of $\\mathcal{S}_q$. In this paper, we show that $\\mathcal{O}_q=\\langle4\\rangle$ if $q=2t+1$ and, $\\mathcal{O}_q=\\langle t\\rangle $ if $q=4t+1$, where $q$ and $t$ are odd primes. This paper also gives a direct method for obtaining the coefficients of irreducible factors of $x^{2^nt}-1$ in $\\ma","authors_text":"Manjit Singh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-27T03:04:30Z","title":"Some subgroups of a finite field and their applications for obtaining explicit factors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.11052","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:c61c34143a465725a5dc6630cbbf13a2c2661fe261b5cf8ba1b499919792b336","target":"record","created_at":"2026-05-18T00:12:07Z","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":"6a9de35ec61b4f1d37c6608b7c170b4e9da13428dee1f0298a39ad73555c2d14","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-27T03:04:30Z","title_canon_sha256":"09b8217c566b74912ac806173ee4bf5b7718f8087f6504f1d9962308188ce681"},"schema_version":"1.0","source":{"id":"1806.11052","kind":"arxiv","version":1}},"canonical_sha256":"8422c62b3b14667d72daeafcde4a4e47743ae0d4dabeaf8a87628eb98535ccf1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8422c62b3b14667d72daeafcde4a4e47743ae0d4dabeaf8a87628eb98535ccf1","first_computed_at":"2026-05-18T00:12:07.144025Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:07.144025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qoq3u9QcQxFzPt9uDiRzw8Go3sk3XL1VsMWtkI6LlPCWSNexkh8/xMhdthnEEzqLc2JSLyel6+uwecXaLL/DAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:07.144627Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.11052","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c61c34143a465725a5dc6630cbbf13a2c2661fe261b5cf8ba1b499919792b336","sha256:f35ba491b7365000e7461328447ec3874a0c9006b55d86dd6797309a5f5774ae"],"state_sha256":"e53089ab5d8c3336b4765f1d15c294018efbf8e2efe865b78ec371a5c901191b"}