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Abstract

In 2001, Bombieri and Zannier studied the Northcott property (N) for infinite Galois extensions of the rationals. In particular they provided a local property of the extensions that imply property (N). Later, Checcoli and Fehm demonstrated the existence of infinite extensions satisfying this local property. In this article, we establish two main results. First, we show that this local property, unlike property (N), is not preserved under finite extensions. Second, we show that, for an infinite Galois extension of Q, such local property cannot be read on the Galois group. More precisely, we exhibit several profinite groups that are realizable over Q by fields that do not satisfy the local property.