Lower Bounds for Heights in Relative Galois Extensions

Shabnam Akhtari, Kevser AktaÅŸ, Kirsti D. Biggs, Alia Hamieh, Kathleen Petersen, Lola Thompson

Research output: Chapter in Book/Report/Conference proceedingChapter

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 � is a number field and �(α) ∕ � 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 �∕ �, depending on the degree of �∕ ℚ and the number of conjugates of α which are multiplicatively independent over �. As a consequence, we obtain a height bound for such α that is independent of the multiplicative independence condition.

Original languageEnglish (US)
Title of host publicationAssociation for Women in Mathematics Series
PublisherSpringer
Pages1-17
Number of pages17
DOIs
StatePublished - 2018
Externally publishedYes

Publication series

NameAssociation for Women in Mathematics Series
Volume11
ISSN (Print)2364-5733
ISSN (Electronic)2364-5741

Bibliographical note

Funding Information:
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.

Funding Information:
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.

Publisher Copyright:
© 2018, The Author(s) and the Association for Women in Mathematics.

Keywords

  • Height of algebraic numbers
  • Lehmer's problem

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