On minimal diameters of algebraic integers in J-fields (Q1061168)

Let \(\alpha\) be an algebraic integer with conjugates \(\alpha_ 1,...,\alpha_ n\). One defines \(diam(\alpha)=\max | \alpha_ i- \alpha_ j|\) and D(\(\alpha)\) as the diameter of the smallest closed disk containing \(\alpha_ 1,...,\alpha_ n\). The author assumes that \(\alpha\) belongs to a J-field (totally real or totally imaginary quadratic extension of a totally real field). He determines all such \(\alpha\) which satisfy \(D(\alpha)<4 \cos (\pi /9)-1=2.758...\) and all such \(\alpha\) which satisfy diam\((\alpha)\leq \sqrt{3} (2 \cos (\pi /9)- 1/2)=2.389....\) He gives a new derivation of a theorem of \textit{R. M. Robinson} [Math. Z. 110, 41-51 (1969; Zbl 0179.077)] that \(diam(\alpha)>3\) if \(\alpha\) is totally real and \(\deg(\alpha)\geq 3.\) In a related paper [Bull. Aust. Math. Soc. 30, 111-121 (1984; Zbl 0545.12001)], the author has considered diam(\(\alpha)\) and D(\(\alpha)\) without the assumption that \(\alpha\) is in a J-field. - There is a misprint in the assumptions of Lemma 7: \(\beta\) should be \(\zeta (1\pm \sqrt{5})/2\).