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 \(\ensuremath {{\mathbb K}}\) is a number field and \(\ensuremath {{\mathbb K}}(\alpha )/\ensuremath {{\mathbb K}}\) 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 \(\ensuremath {{\mathbb {F}}}/\ensuremath {{\mathbb K}}\), depending on the degree of \(\ensuremath {{\mathbb K}}/\ensuremath {\mathbb {Q}}\) and the number of conjugates of α which are multiplicatively independent over \(\ensuremath {{\mathbb K}}\). As a consequence, we obtain a height bound for such α that is independent of the multiplicative independence condition.
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Acknowledgements
This work began as a research project for the working group Heights of Algebraic Integers at the Women in Numbers Europe 2 workshop held at the Lorentz Center at the University of Leiden. The authors would like to thank the organizers of the workshop and the Lorentz Center for their hospitality.
Research of Shabnam Akhtari is supported by the NSF grant DMS-1601837. Kirsti Biggs is supported by an EPSRC Doctoral Training Partnership. Research of Alia Hamieh is partially supported by a PIMS postdoctoral fellowship. Research of Kathleen Petersen is supported by Simons Foundation Collaboration grant number 209226 and 430077; she would like to thank the Tata Institute of Fundamental Research for their hospitality while preparing this manuscript. Lola Thompson is supported by an AMS Simons Travel Grant, by a Max Planck Institute fellowship during the Fall 2016 semester, and by the NSF grant DMS-1440140 while in residence at the Mathematical Sciences Research Institute during the Spring 2017 semester.
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Akhtari, S., Aktaş, K., Biggs, K.D., Hamieh, A., Petersen, K., Thompson, L. (2018). Lower Bounds for Heights in Relative Galois Extensions. In: Bouw, I., Ozman, E., Johnson-Leung, J., Newton, R. (eds) Women in Numbers Europe II. Association for Women in Mathematics Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-74998-3_1
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