Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Chinese Remainder Theorem for Cyclotomic Polynomials in $\mathbf{Z}[X]$   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 any $\epsilon > 0$
to determine a basis of Smith vectors (see Definition 3.1) for the codomain of
$\Psi_n$. In general when $R$ is an integral domain, we prove that the
determinant of $\Psi : R[X]/(\prod_j f_j) \to \bigoplus_j R[X]/(f_j)$ written
with respect to the standard basis is $\prod_{1 \leqslant i < j
  \leqslant n} \mathcal{R}(f_j, f_i)$, where $f_i$'s are pairwise relatively
prime monic polynomials and $\mathcal{R}(f_j, f_i)$ is the resultant of $f_j$
and $f_i$.

25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: stat
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
3
The categories of this paper are: physics
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ex