Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4584279 | Journal of Algebra | 2015 | 40 Pages |
Abstract
By the Chinese remainder theorem, the canonical mapΨn:R[X]/(Xnâ1)ââd|nR[X]/Φd(X) is an isomorphism when R is a field whose characteristic does not divide n and Φd is the dth cyclotomic polynomial. When R is the ring 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 Ψn (when R=Z) using the prime factorisation of n. We also give a pictorial algorithm using Young tableaux that takes O(n3+ϵ) bit operations for any ϵ>0 to determine a basis of Smith vectors (see Definition 3.1) for Ψn. In general when R is an integral domain, we prove that the determinant of the matrix of Ψ:R[X]/(âjfj)ââ¨jR[X]/(fj) written with respect to the standard basis is â1⩽i
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kamalakshya Mahatab, Kannappan Sampath,