  We start with a brief survey on the Northcott property for subfields of the
algebraic numbers $\Qbar$. Then we introduce a new criterion for its validity
(refining the author's previous criterion), addressing a problem of Bombieri.
We show that Bombieri and Zannier's theorem, stating that the maximal abelian
extension of a number field $K$ contained in $K^{(d)}$ has the Northcott
property, follows very easily from this refined criterion. Here $K^{(d)}$
denotes the composite field of all extensions of $K$ of degree at most $d$.
