Parity of the number of irreducible factors for composite polynomials (Q973959)

The focus of the paper is on the parity of the number of irreducible factors of a given polynomial over a finite field. Here the authors consider the discriminants of the composition of some polynomials over a finite field. They establish a relation between the discriminant of the composed one and the original one. They apply this then to obtain results like the following. For any integer \(k>3\), \(k>l\) and \(l\geq 1\), the pentanomial \(f(x)=x^{2^k-1}+x^{2^l+1}+x^{2^l}+x+1\) has always an odd number of irreducible factors over the finite field with two elements.