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
