Factorization of a class of polynomials over finite fields

We study the factorization of polynomials of the form Fr(x)=bxqr+1−axqr+dx−c over the finite field Fq. We show that these polynomials are closely related to a natural action of the projective linear group PGL(2,q) on non-linear irreducible polynomials over Fq. Namely, irreducible factors of Fr(x) are exactly those polynomials that are invariant under the action of some non-trivial element [A]∈PGL(2,q). This connection enables us to enumerate irreducibles which are invariant under [A]. Since the class of polynomials Fr(x) includes some interesting polynomials like xqr−x or xqr+1−1, our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over Fq. At the same time, we generalize recent results about certain invariant polynomials over the binary field F2.