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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.

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