Lower Bounds for Heights in Relative Galois Extensions

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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 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.

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Akhtari, S., AktaÅŸ, K., Biggs, K. D., Hamieh, A., Petersen, K., & Thompson, L. (2018). Lower Bounds for Heights in Relative Galois Extensions. In Association for Women in Mathematics Series (Vol. 11, pp. 1–17). Springer. https://doi.org/10.1007/978-3-319-74998-3_1

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