Algebraic numbers with elements of small height. (English) Zbl 1503.11151
Summary: 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 \(\alpha>1\) is called a Salem number if \(\alpha\) and \(1/\alpha\) are Galois conjugate and all other Galois conjugates of \(\alpha\) 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.
MSC:
| 11U09 | Model theory (number-theoretic aspects) |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03C60 | Model-theoretic algebra |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
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