Dickson curves (Q885589)

Summary: Let \(\mathbb F_q\) denote the finite field of order \(q\) and odd characteristic \(p\). For \(a\in \mathbb F_q\), let \(g_d(x,a)\) denote the Dickson polynomial of degree \(d\) defined by \[ g_d(x,a)=\sum_{i=0}^{[d/2]}d/(d-i)\binom{d-i}i(-a)^ix^{d-2i}. \] Let \(f(x)\) denote a monic polynomial with coefficients in \(\mathbb F_q\). Assume that \(f^2(x)-4\) is not a perfect square and \(\gcd(p,d)=1\). Also assume that \(f(x)\) and \(g_2(f(x),1)\) are not of the form \(g_d(h(x),c)\). In this note, we show that the polynomial \(g_d(y,1)-f(x)\in \mathbb F_q[x,y]\) is absolutely irreducible.