# Lower Bounds for Heights in Relative Galois Extensions

## TL;DR

This paper derives explicit lower bounds for the height of algebraic numbers in Galois extensions, extending previous results and providing bounds that depend on field degrees and conjugate properties.

## Contribution

It extends height lower bounds to relative Galois extensions, offering effective bounds depending on field degrees and multiplicative independence.

## Key findings

- Effective height bounds for algebraic numbers in Galois extensions.
- Explicit bounds depending on degree and conjugates.
- Height bounds independent of multiplicative independence.

## 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 non-zero 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.