Equal-degree factorization of binomials and trinomials over finite fields

Let \(n\) $n$ be a positive integer and \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ , a finite field of \(q\) $q$ elements such that the product of distinct prime factors of \(n\) $n$ divides \(q-1\) $q-1$ . Equal-degree factorization (EDF) factors a polynomial whose irreducible factors have the same degree. In this paper, for any \(\alpha \in {\mathbb {F}}_q^*\) $\alpha \in {\mathbb{F}}_q^{\ast }$ , we characterize all irreducible factors of \(x^n-\alpha \) $x^n-\alpha$ over \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ using the equal-degree factorization phenomena of binomials and trinomials over \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ under some conditions on \(n, q\) $n,q$ , and \(\alpha \) $\alpha$ .