Let n be any positive integer and \(\mathbb {F}_q\) be a finite field with q elements, where q is a prime power. In this paper, we give the irreducible factorization of the n-th cyclotomic polynomial \(\Phi _n(x)\) over \(\mathbb {F}_q\) , which can be achieved by utilizing the irreducible factorization of lower-order cyclotomic polynomial \(\Phi _m(x)\) over \(\mathbb {F}_q\) when \(\text {rad}(n)|(q+1)\) and \(m=\gcd (n,q+1)>1\) . By doing so, we provide a unified explanation of the irreducible factorization of \(\Phi _n(x)\) over \(\mathbb {F}_q\) and elucidate the mathematical mechanism for computing the coefficients of these irreducible factors. When \(4\not \mid n\) , there is a one-to-one correspondence between the irreducible factors of \(\Phi _m(x)\) and \(\Phi _n(x)\) over \(\mathbb {F}_q\) . In addition, some numerical examples are presented to illustrate this concept.
