Factorization of a class of polynomials over finite fields

Abstract

We study the factorization of polynomials of the form F r ( x ) = b x q r + 1 − a x q r + d x − c over the finite field F q . 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 F q . Namely, irreducible factors of F r ( 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 F r ( x ) includes some interesting polynomials like x q r − x or x q r + 1 − 1 , our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over F q . At the same time, we generalize recent results about certain invariant polynomials over the binary field F 2 .