Monogenity of iterates of irreducible binomials (Q6600738)

A monic polynomial \(f(x) \in {\mathbb Z}[x]\) is called monogenic if \(f(x)\) is irreducible over \(\mathbb Q\) and for any root \(\theta\) of \(f(x)\) the ring of integers of the field \({\mathbb Q}(\theta)\) is \({\mathbb Z}[\theta]\). Let \(f(x)=x^m-a\), where \(a \in {\mathbb Z}\), be a monic irreducible polynomial. Consider the \(n\)th iterate of \(f(x)\), where \(f^{n}(x)=f(f^{n-1}(x))\). For each \(n \geq 1\) let \(\alpha_n\) be an algebraic integer satisfying \(f^n(\alpha_n)=0\), \(K_n={\mathbb Q}(\alpha_n)\), and let \({\mathcal O}_{K_n}\) be the ring of integers of the field \(K_n\). The authors prove that then \({\mathcal O}_{K_n}={\mathbb Z}[\alpha_n]\) for all \(n \leq N\) if and only if \(p^2\) does not divide \(a^p-a\) for each prime \(p\) dividing \(m\) and \(f^n(0)\) is squarefree for all \(n \leq N\). This result implies that the iterates of \(f(x)=x^2-2\) are all monogenic. They conclude with the open question asking if for \(m \geq 3\) there is \(a \in {\mathbb Z}\) such that the iterates of \(f(x)=x^m-a\) are all monogenic.