A note on algebraic integers (Q2533229)

Let \(\alpha\) be an algebraic integer and \(\alpha_j\) \((j=1,\ldots,n)\). The paper is concerned with estimating \(\max_{1\le j\le n} \vert\alpha_j\vert\) and contains several refinements of the previous result in that direction due to \textit{J. W. S. Cassels} [J. Math. Sci. 1, 1--8 (1966; Zbl 0147.30803)], \textit{H. Zassenhaus} and the reviewer [Mich. Math. J. 12, 81--85 (1965; Zbl 0128.03402)]. Since the author has since improved some of his results (to be published) only one of them is quoted. Let \(s\) be the number of complex conjugate pairs of roots. If \(2s \le n (\log10n - \log 2.1)/\log10n\) then \[ \max_{1\le j\le n} \vert\alpha_j\vert > 1 +1/10 n. \]