Dickson curves

Abstract

Let kq denote the finite field of order q and odd characteristic p. For a∈kq, let gd(x,a) denote the Dickson polynomial of degree d defined by gd(x,a)=∑i=0[d/2]d/(d−i)(d−ii)(−a)ixd−2i. Let f(x) denote a monic polynomial with coefficients in kq. Assume that f2(x)−4 is not a perfect square and gcd⁡(p,d)=1. Also assume that f(x) and g2(f(x),1) are not of the form gd(h(x),c). In this note, we show that the polynomial gd(y,1)−f(x)∈kq[x,y] is absolutely irreducible.