On Widmer's criteria for the Northcott property (Q6621274)

A subset \(S\) of algebraic number \(\overline{\mathbb{Q}}\) is said to have Northcott property if for every \(T > 0\) \[\N \{ \alpha \in S \,\, | \,\, h(\alpha) < T\} \N\]\Nis a finite set, where \(h\) is the absolute Weil height. It is an important theme in Diophantine geometry to study infinite extensions of \(\mathbb{Q}\) satisfying property.\N\NFor \(M / K\) a finite extension of number fields, let \(D_{M / K}\) denote the relative discriminant and \(N_{M / K}\) the relative norm, and \N\[\N\gamma(M / K)=\sup _{K \subseteq F, [F : K] < \infty}\left(N_{F / Q}\left(D_{M F / F}\right)\right)^{1 /[[M F: \mathrm{Q}][M F: F])}.\tag{1}\N\]\NRecently, \textit{M. Widmer} [Essent. Number Theory 2, No. 1, 1--14 (2023; Zbl 1548.11105)], refining his earlier criterion, have shown that if for an infinite algebraic extension \(L\) of a number field \(K\) \N\[\N\liminf _{ \substack{K \subset M \subset L, \\\N[M:K] < \infty}} \gamma(M / K)=\infty, \N\]\Nthen \(L\) satisfies Northcott property.\N\NIn this paper, the authors have shown that for Galois extensions \(M/K\), restricting the supremum over \(F \subseteq M\) in (1) does not change \(\gamma(M/K)\) and thus simplifying Widmer's criteria. Using this simplification in Widmer's criteria, they constructed infinite extension having Northcott property but not satisfying Widmer's criteria.