Explicit factorization of X2mpn−1X2mpn−1 over a finite field

Let FqFq be a finite field of odd order q   and m,nm,n be positive integers. In this paper, the irreducible factorization of X2mpn−1X2mpn−1 over FqFq is given in a very explicit form, where p   is an odd prime divisor of q−1q−1. It is shown that all the irreducible factors of X2mpn−1X2mpn−1 over FqFq are either binomials or trinomials.