$\mathbf{Z}[X]$中分圆多项式的孙子剩余定理
Chinese Remainder Theorem for Cyclotomic Polynomials in $\mathbf{Z}[X]$
摘要：
利用孙子剩余定理，证明了当$R$是特征不整除$n$的域，$Φ_d$是$d$次分圆多项式时，规范映射\[Ψ_n: R[X]/(X^n-1)\to \oplus_{d|n} R[X]/Φ_d(X)\]是同构的.当$R$是有理数环$\mathbf{Z}$时，这个映射是单射的但不是满射的。本文利用$n$的素因子分解给出了$Ψ_n$（当$R=\mathbb{Z}$）的上核的初等因子的显式表达式.我们还给出了一个图形算法，使用杨表，采取$O(n^{3+ε})$位操作的任何$ε> 0$，以确定史密斯矢量的基础（见定义3.1）的$Ψ_n$的余域。在一般情况下，当$R$是整环时，我们证明了$Ψ: R[X]/(\prod_j f_j) \to \bigoplus_j R[X]/(f_j)$关于标准基的行列式是$\prod_{1 \leqslant i < j \leqslant n} \mathcal{R}(f_j, f_i)$，其中$f_i$是两两相对素一元多项式，$\mathcal{R}(f_j, f_i)$是$f_j$和$f_i$的结式.
英文摘要：
By the Chinese remainder theorem, the canonical map \[Ψ_n: R[X]/(X^n-1)\to \oplus_{d|n} R[X]/Φ_d(X)\] is an isomorphism when $R$ is a field whose characteristic does not divide $n$ and $Φ_d$ 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 $Ψ_n$(when $R=\mathbb{Z}$) using the prime factorisation of $n$. We also give a pictorial algorithm using Young Tableaux that takes $O(n^{3+ε})$ bit operations for any $ε> 0$ to determine a basis of Smith vectors (see Definition 3.1) for the codomain of $Ψ_n$. In general when $R$ is an integral domain, we prove that the determinant of $Ψ: R[X]/(\prod_j f_j) \to \bigoplus_j R[X]/(f_j)$ 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 $f_i$.
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