Constructing totally $p$-adic numbers of small height

TL;DR

This paper generalizes Bombieri and Zannier's construction of algebraic numbers with small height to maximal Galois extensions of number fields with specified local properties, providing explicit bounds.

Contribution

It extends the effective construction of small-height algebraic numbers from rationals to general number fields with prescribed local conditions.

Findings

Provides explicit upper bounds for the lim inf of heights in these fields

Generalizes previous results from rationals to arbitrary number fields

Offers an effective method for constructing totally $p$-adic numbers of small height

Abstract

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behaviour at finitely many places.