Factorization of Q(h(T)(x)) over a finite field, where Q(x) is irreducible and h(T)(x) is linear II

Abstract

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(qi)[x], and let H(x) = h(T)(x) be a linear polynomial in GF(q)[x]. We give the degrees of the irreducible factors of Q(H(x)) in GF(qi)[x], and the number of irreducible factors of each degree. We consider the special cases when H(x) is a trace function, and when h(x) is cyclotomic. Finally, we give several examples.