The elementary theory of large fields of totally S-adic numbers

We analyze the elementary theory of certain fields of totally S-adic algebraic numbers that were introduced and studied by Geyer-Jarden and Haran-Jarden-Pop. In particular, we provide an axiomatization of these theories and prove their decidability, thereby giving a common generalization of classical decidability results of Jarden-Kiehne, Fried-Haran-V"olklein and Ershov.