{"paper":{"title":"Chinese Remainder Theorem for Cyclotomic Polynomials in $\\mathbf{Z}[X]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.RA"],"primary_cat":"math.NT","authors_text":"Kamalakshya Mahatab, Kannappan Sampath","submitted_at":"2014-01-29T23:09:20Z","abstract_excerpt":"By the Chinese remainder theorem, the canonical map \\[\\Psi_n: R[X]/(X^n-1)\\to \\oplus_{d|n} R[X]/\\Phi_d(X)\\] is an isomorphism when $R$ is a field whose characteristic does not divide $n$ and $\\Phi_d$ is the $d$th cyclotomic polynomial. When $R$ is the ring $\\mathbf{Z}$ of rational integers, this map is injective but not surjective. In this paper, we give an explicit formula for the elementary divisors of the cokernel of $\\Psi_n$(when $R=\\mathbb{Z}$) using the prime factorisation of $n$. We also give a pictorial algorithm using Young Tableaux that takes $O(n^{3+\\epsilon})$ bit operations for an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7696","kind":"arxiv","version":3},"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"}