3 problems

Let $K$ be a number field and let $f$ be a quadratic polynomial over $K$. Write $T_{\infty}(f,\alpha)=\{f^{-N}(\alpha)\}_{N\in\mathbb{Z}_{\geq 0}}$ for the tree of iterated preimag…

Let $G_N(f,\alpha)=\operatorname{Gal}(K_N(f,\alpha)/K)$ denote the finite-level arboreal Galois group, where $K_N(f,\alpha)$ is the splitting field of $f^N-\alpha$. For the polynom…

Let $f$ be a polynomial over a global field $K$. For $alpha$ in $K$ and $N$ in $\mathbb{Z}_{\geq 0}$, let $K_N(f,\alpha)=K(f^{-N}(\alpha))$ be the arboreal field and let…