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 language | English (US) |
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Title of host publication | Association for Women in Mathematics Series |
Publisher | Springer |
Pages | 1-17 |
Number of pages | 17 |
DOIs | |
State | Published - 2018 |
Externally published | Yes |
Publication series
Name | Association for Women in Mathematics Series |
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Volume | 11 |
ISSN (Print) | 2364-5733 |
ISSN (Electronic) | 2364-5741 |
Bibliographical note
Publisher Copyright:© 2018, The Author(s) and the Association for Women in Mathematics.
Keywords
- Height of algebraic numbers
- Lehmer's problem