On the Northcott property and other properties related to polynomial mappings

We prove that if $K/\mathbb{Q}$ is a Galois extension of finite exponent and $K^{(d)}$ is the compositum of all extensions of $K$ of degree at most $d$, then $K^{(d)}$ has the Bogomolov property and the maximal abelian subextension of $K^{(d)}/\mathbb{Q}$ has the Northcott property. Moreover, we prove that given any sequence of finite solvable groups $\{G_m\}_m$ there exists a sequence of Galois extensions $\{K_m\}_m$ with $\text{Gal}(K_m/\mathbb{Q})=G_m$ such that the compositum of the fields $K_m$ has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in $\mathbb{Q}^{(d)}$. We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.