  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.
