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The Fekete-Szegő theorem with splitting conditions. I. (English) Zbl 0946.11025
It 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 $$\mathbb{Q}_p$$.

##### MSC:
 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11R04 Algebraic numbers; rings of algebraic integers 31C15 Potentials and capacities on other spaces 11S05 Polynomials
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