Iterates of prime producing polynomials and their Galois groups

TL;DR

This paper proves that certain polynomial specializations over finite fields produce prime elements and satisfy Odoni's conjecture, with implications for Galois groups and prime generation.

Contribution

It establishes conditions under which polynomial specializations over finite fields generate primes and confirms Odoni's conjecture for these polynomials.

Findings

Polynomial specializations produce primes over large finite fields

The Galois group associated with these polynomials is surjective

Supports Odoni's conjecture in this context

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 $\phi(t)$ in the variable $t$, the polynomial $F(t,\phi)$ is a prime producing polynomial over large finite fields under suitable irreducible specialization. We also prove that $F(t,\phi)$ satisfies Odoni's conjecture, namely the arboreal Galois representation associated to $F(t,\phi)$ is surjective.