论文标题
有限字段（ii）的$ x^n-1 $进一步分解
Further factorization of $x^n-1$ over finite fields (II)
论文作者
论文摘要
令$ \ bbb f_q $为有$ q $元素的有限字段。让$ n $是一个积极的整数，带有Radical $ rad（n）$，即，$ n $的独特主要除数的产物。如果$ q $ modulo $ rad（n）$的订单是1或素数，则不可约束的因素和不可约束的不可约束$ x^n-1 $的计数公式，由$ \ bbb f_q $获得。 Fields Appl 54：197-215，2018）。在本文中，我们将$ x^{n} -1 $显式分配到$ \ bbb f_q [x] $中的不可约因素中，并计算出不可减少因素的数量时，当$ q $ q $ modulo $ rad（n）$的订单是两个primes的产品。
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.
