On the reducibility of some composite polynomials over finite fields

Let Fq be a Galois field with q = p elements of characteristic p, and Fq is multiplicative group of Fq. The problem of irreducibility of polynomials and determining the reducibility of a given polynomial stems both from mathematical theory and applications. At mathematical aspects determining the reducibility of a polynomial often appears in number theory, combinatorics and algebraic geometries. The study of irreducible polynomials is an old but currently still active subject. One of the methods for constructing irreducible polynomials is composition method. Probably the most powerful result in this area is the following theorem by S. Cohen that states as follows.