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Conference Paper

#### Lower bounds for heights in relative Galois extensions

##### External Resource

https://doi.org/10.1007/978-3-319-74998-3_1

(Publisher version)

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##### Fulltext (public)

arXiv:1704.02995.pdf

(Preprint), 196KB

##### Supplementary Material (public)

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##### Citation

Akhtari, S., Aktaş, K., Biggs, K., Hamieh, A., Petersen, K., & Thompson, L. (2018).
Lower bounds for heights in relative Galois extensions. In *Women in Numbers Europe II: Contributions
to Number Theory and Arithmetic Geometry* (pp. 1-17). Cham: Springer.

Cite as: https://hdl.handle.net/21.11116/0000-0003-D9A9-A

##### 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.

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.