We start with a brief survey on the Northcott property for subfields of the algebraic
numbers
.
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
contained
in
has
the Northcott property, follows very easily from this refined criterion. Here
denotes the composite
field of all extensions of
of degree at most
.