Lower Bounds for Heights in Relative Galois Extensions

Abstract

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 𝕂 is a number field and 𝕂(α) ∕ 𝕂 is Galois. Our second result establishes an explicit height bound for any nonzero element α which is not a root of unity in a Galois extension 𝔽∕ 𝕂, depending on the degree of 𝕂∕ ℚ and the number of conjugates of α which are multiplicatively independent over 𝕂. As a consequence, we obtain a height bound for such α that is independent of the multiplicative independence condition.