# On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop

• ### Lior Bary-Soroker

Tel Aviv University, Israel
• ### Arno Fehm

University of Konstanz, Germany

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## Abstract

Given a finite set $\mathfrak{S}$ of places of a number field, we prove that the field of totally $\mathfrak{S}$-adic algebraic numbers is not Hilbertian.