Dynamical Galois groups of trinomials and Odoni's conjecture

We prove Odoni’s conjecture in all prime degrees; namely, we prove that for every positive prime $p$, there exists a degree $p$ polynomial $\phi \in \mathbb{Z}[x]$ with surjective arboreal Galois representation. We also show that Vojta’s conjecture implies the existence of such a polynomial in many degrees $d$ which are not prime.