{"paper":{"title":"Factoring polynomials of the form $f(x^n)\\in \\mathbb{F}_q[x]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"F.E. Brochero Mart\\'inez, Lucas Reis","submitted_at":"2015-11-28T18:35:05Z","abstract_excerpt":"Let $f(x)\\in \\mathbb{F}_q[x]$ be an irreducible polynomial of degree $m$ and exponent $e$, and $n$ be a positive integer such that $\\nu_p(q-1)\\ge \\nu_{p}(e)+\\nu_p(n)$ for all $p$ prime divisor of $n$. We show a fast algorithm to determine the irreducible factors of $f(x^n)$. We also show the irreducible factors in the case when ${\\rm rad}(n)$ divides $q-1$ and ${\\rm gcd}(m, n)=1$. Finally, using this algorithm we split $x^n-1$ into irreducible factors, in the case when $n=2^mp^t$ and $q$ is a generator of the group $\\mathbb{Z}_{p^2}^*$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08918","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"}