Polynomials with Surjective Arboreal Galois Representations Exist in Every Degree

Let~\(E\) be a Hilbertian field of characteristic~\(0\). R.W.K. Odoni conjectured that for every positive integer~\(n\) there exists a polynomial~\(f\in E[X]\) of degree~\(n\) such that each iterate~\(f^{\circ{k}}\) of~\(f\) is irreducible and the Galois group of the splitting field of~\(f^{\circ k}\) is isomorphic to the automorphism group of a regular,~\(n\)-branching tree of height~\(k.\) We prove this conjecture when~\(E\) is a number field.