Abstract
AbstractWe define a new height function $\mathcal{R}(\alpha)$, the Remak height of an algebraic number $\alpha$. We give sharp upper and lower bounds for $\mathcal{R}(\alpha)$ in terms of the classical Mahler measure $M(\alpha)$. Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm $\pm 1$ lying on two circles centred at 0. We give a complete characterization of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalization of Salem numbers which we call extended Salem numbers.AMS 2000 Mathematics subject classification: Primary 11R06
Publisher
Cambridge University Press (CUP)
Cited by
14 articles.
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