On certain infinite extensions of the rationals with Northcott property

Abstract

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott's Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that $\IQ_{ab}^{(d)}$, the maximal abelian subfield of the field generated by all algebraic numbers of degree at most $d$, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. $\IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...)$ has the Northcott property if and only if $2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$ tends to infinity

On fields of algebraic numbers with bounded local degrees

“…These constructions are interesting for two reasons: firstly they permit to avoid the use of Shafarevich's result. Secondly, coming back to one of the original motivations of this work, Widmer proves in paper [16] that the Northcott property for an infinite algebraic extension K of Q is strictly related to the behavior of the discriminants of certain finite subextensions of K. Therefore, knowing how to concretely construct fields described in Chapter 4 could be a step towards understanding whether these extensions might have the Northcott property or not.…”

Heights and totally p-adic numbers

“…This, however, is not true as was shown ineffectively by Kubota and Liardet [19]. Effective constructions of counterexamples were presented in [11] and [27]. Our argument leads to another class of counterexamples, which are also presented in Section 4.…”

Relative Bogomolov extensions

“…Our main tool is the lower bound [19,Theorem 2], due to Silverman. This bound is written in notation more similar to ours in [20,Section 3], where the author uses it effectively to give examples of fields satisfying the closely related Northcott property, (N). This stronger property, first defined along with (B) in [7], is satisfied by a field K if for any T at most finitely points in K have height at most T .…”

“…Widmer exploited the dependence only on relative ramification in this bound to produce a ramification criterion for the Northcott property [20]. Our most general criterion for the relative Bogomolov property using this bound is the following theorem.…”