Discriminants and the irreducibility of a class of polynomials in a finite field of arbitrary characteristic

AbstractThere has been some interest in finding irreducible polynomials of the type f(A(x)) for certain classes of linearized polynomial A(x) over a finite field GF(pm). The main result of this paper proves the stronger result that there are no further irrducible cases of f(A(x)) for an extended class that contains that of linearized polynomials, but for p ≠ 2. (The case p = 2 we have considered in [O. Moreno, Discriminants and the irreducibility of a class of polynomials, “Lecture Notes in Computer Science,” Vol. 228, Proc. 2nd Int. Conf. AAECL-2, pp. 178–181]). In order to reach this result, and also of independent interest, the discriminant and the parity of the factors of polynomials f(A(x)) are computed. Also a new proof of a result first established in [S. Agou, Irréductibilité des polynômes f(Σi=0maiXpi) sur un corp fini Fpx, Canad. Math. Bull. 23 (1980), 207–212] is given