Iterates of prime producing polynomials and their Galois groups

Abstract

Let $\F_q$ be a finite field of characteristic $p>0$. We prove that, given $F(t,x)\in \F_q[t][x]$ an irreducible separable monic polynomial in the variable $x$ and a generic monic polynomial $\varphi (t)$ in the variable $t$, the polynomial $F(t,\varphi )$ is a prime producing polynomial over large finite fields under suitable irreducible specialization. We also prove that $F(t,\varphi )$ satisfies Odoni's conjecture, namely the arboreal Galois representation associated to $F(t,\varphi )$ is surjective.

arXiv comment18 pages