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Algebraic numbers with elements of small height
Author(s) -
Göral Haydar
Publication year - 2019
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201700043
Subject(s) - mathematics , conjecture , algebraic extension , algebraic number , simple (philosophy) , discrete mathematics , combinatorics , pure mathematics , mathematical analysis , differential algebraic equation , ordinary differential equation , philosophy , epistemology , differential equation
In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer α > 1 is called a Salem number if α and 1 / α are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers.