Parity of the number of irreducible factors for composite polynomials

Various results on the parity of the number of irreducible factors of given polynomials over finite fields have been obtained in the recent literature. Those are mainly based on Stickelberger's and Swan's theorem in which discriminants of polynomials over a finite field or the integral ring Z play an important role. In this paper we consider discriminants of the composition of some polynomials over finite fields. A relation between the discriminants of the composed polynomial and the original ones will be established. We apply this to obtain some results concerning the parity of the number of irreducible factors for several special types of polynomials over finite fields.