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

On fields of totally $S$ -adic numbers. With an appendix by Florian Pop

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

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