zbMATH — the first resource for mathematics

On the size of algebraic integers. (Russian) Zbl 0732.11052
For an algebraic integer \(\alpha\) denote by \(\alpha '\) the maximum of the absolute values of conjugates of \(\alpha\). It follows from a result of E. Dobrowolski [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 26, 291-292 (1978; Zbl 0393.12003)] that if \(f(n)=\exp (\frac{\log n}{6n^ 2})\) and \(\alpha '<f(n)\) where n denotes the degree of \(\alpha\) then \(\alpha\) is a root of unity. A later result of E. Dobrowolski [Acta Arith. 34, 391-401 (1979; Zbl 0416.12001)] shows that one can replace here f(n) by \(\exp ((1-\epsilon)\frac{(\log \log n \log n)^ 3}{n})\) for \(n>n_ 0(\epsilon)\). The author improves slightly the first of these results and points out that although his result is weaker than the second it is effective.
Note of reviewer: It is stated in the second paper of Dobrowolski that if one replaces in his theorem 1-\(\epsilon\) by 1/1200 then its assertion holds for all n, a much stronger result than that in the paper reviewed.

11R04 Algebraic numbers; rings of algebraic integers
11R09 Polynomials (irreducibility, etc.)