Dynamical Galois groups of trinomials and Odoni's conjecture

We prove that for every prime p, there exists a degree p polynomial whose arboreal Galois representation is surjective, that is, whose iterates have Galois groups over Q that are as large as possible subject to a natural constraint coming from iteration. This resolves in the case of prime degree a conjecture of Odoni from 1985. We also show that a standard height uniformity conjecture in arithmetic geometry implies the existence of such a polynomial in many degrees d which are not prime.