2017/04/10 by Akhtari, Shabnam, Aktaş, Kevser, Biggs, Kirsti +3
#11G50 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1704.02995
The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an algebraic number α when the base field \mathbbK is a number field and \mathbbK(α)/\mathbbK is Galois. Our second result establishes an explicit height bound for any non-zero element α which is not a root of unity in a Galois extension \mathbbF/\mathbbK, depending on the degree of \mathbbK/ℚ and the number of conjugates of α which are multiplicatively independent over \mathbbK. As a consequence, we obtain a height bound for such α that is independent of the multiplicative independence condition.