The Fekete-Szegő theorem with splitting conditions. I.

*(English)*Zbl 0946.11025It has been shown by M. Fekete and G. Szegő [Math. Z. 63, 158-172 (1955; Zbl 0066.27002)] that one can find infinitely many full sets \(C\) of conjugated algebraic integers in every neighbourhood of a compact set in the complex plane, which is stable under conjugation and whose logarithmic capacity is \(\geq 1\). If \(E\) is real then, as shown by R. M. Robinson [Math. Z. 84, 415-427 (1964; Zbl 0126.02902)], one can obtain such sets \(C\) consisting of totally real algebraic integers. The author obtains now an extension of these results, which takes care of the \(p\)-adic valuations: let \(T\) be a finite set of primes of the rational field (which may include also the infinite prime), let \(r>\prod_{p\in T}p^{1/(p-1)}\) and denote by \(I_r\) the interval \([-2r,2r]\). Then there exist infinitely many algebraic integers all of whose conjugates in the complex field lie in \(I_r\) and all of whose conjugates in the algebraic closure of the \(p\)-adic field \(\mathbb{Q}_p\) lie in the ring \(\mathbb{Z}_p\) of integers of \(\mathbb{Q}_p\).

Reviewer: Wladyslaw Narkiewicz (Wrocław)