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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"},"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":"1806.11052","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-27T03:04:30Z","cross_cats_sorted":[],"title_canon_sha256":"09b8217c566b74912ac806173ee4bf5b7718f8087f6504f1d9962308188ce681","abstract_canon_sha256":"6a9de35ec61b4f1d37c6608b7c170b4e9da13428dee1f0298a39ad73555c2d14"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:07.144627Z","signature_b64":"Qoq3u9QcQxFzPt9uDiRzw8Go3sk3XL1VsMWtkI6LlPCWSNexkh8/xMhdthnEEzqLc2JSLyel6+uwecXaLL/DAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8422c62b3b14667d72daeafcde4a4e47743ae0d4dabeaf8a87628eb98535ccf1","last_reissued_at":"2026-05-18T00:12:07.144025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:07.144025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some subgroups of a finite field and their applications for obtaining explicit factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Manjit Singh","submitted_at":"2018-06-27T03:04:30Z","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. 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