有限域上$x^n-1$的进一步分解（II）
Further factorization of $x^n-1$ over finite fields (II)
摘要：
设$\Bbb F_q$是具有$q$元素的有限域。设$n$是一个带根式$rad(n)$的正整数，即$n$的不同素因子的乘积。如果$q$模$rad(n)$的阶数是1或素数，则Martínez，Vergara和Oliveira（Des Codes Cryptogr 77（1）：277-286，2015）以及Wu，Yue和Fan（Finite Fields Appl 54：197-215，2018）获得了$\Bbb F_q$上$x^n-1$的不可约因子分解和不可约因子的计数公式。本文将$x^{n}-1$显式分解为$\Bbb F_q[x]$中的不可约因子，并计算了当$q$模$rad(n)$的阶为两个素数的乘积时不可约因子的个数.
英文摘要：
Let $\Bbb F_q$ be a finite field with $q$ elements. Let $n$ be a positive integer with radical $rad(n)$, namely, the product of distinct prime divisors of $n$. If the order of $q$ modulo $rad(n)$ is either 1 or a prime, then the irreducible factorization and a counting formula of irreducible factors of $x^n-1$ over $\Bbb F_q$ were obtained by Mart\'ınez, Vergara, and Oliveira (Des Codes Cryptogr 77 (1) : 277-286, 2015) and Wu, Yue, and Fan (Finite Fields Appl 54: 197-215, 2018). In this paper, we explicitly factorize $x^{n}-1$ into irreducible factors in $\Bbb F_q[x]$ and calculate the number of the irreducible factors when the order of $q$ modulo $rad(n)$ is a product of two primes.
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