  Recently, Widmer introduced a new sufficient criterion for the Northcott
property on the finiteness of elements of bounded height in infinite algebraic
extensions of number fields. We provide a simplification of Widmer's criterion
when the extension is abelian, and use this to exhibit fields with the
Northcott property that do not satisfy Widmer's new criterion. We also show how
the construction of pseudo algebraically closed fields with the Northcott
property, carried out in previous work using the first version of Widmer's
criterion, can be simplified.
