Irreducible polynomials from a cubic transformation

Abstract

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot f\bigl(R(x)\bigr)$ for some $f(x)\in\mathbb{F}_q[x]$. As an application, we recover the number of irreducible transformation shift registers of order three, previously computed by Jiang and Yang.