{"paper":{"title":"Further factorization of $x^n-1$ over a finite field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT","math.NT"],"primary_cat":"cs.IT","authors_text":"Qin Yue, Shuqin Fan, Yansheng Wu","submitted_at":"2017-10-22T14:03:09Z","abstract_excerpt":"Let $\\Bbb F_q$ be a finite field with $q$ elements and $n$ a positive integer. Mart\\'inez, Vergara and Oliveira \\cite{MVO} explicitly factorized $x^{n} - 1$ over $\\Bbb F_q$ under the condition of $rad(n)|(q-1)$. In this paper, suppose that $rad(n)\\nmid (q-1)$ and $rad(n)|(q^w-1)$, where $w$ is a prime, we explicitly factorize $x^{n}-1$ into irreducible factors in $\\Bbb F_q[x]$ and count the number of its irreducible factors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07943","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"}