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