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totally s-adic numbers

Journal of the Institute of Mathematics of Jussieu (1)

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THE ELEMENTARY THEORY OF LARGE FIELDS OF TOTALLY $\mathfrak{S}$ -ADIC NUMBERS
Arno Fehm
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 16 / Issue 1 / February 2017

Published online by Cambridge University Press:

23 April 2015, pp. 121-154

Print publication:

February 2017
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We analyze the elementary theory of certain fields $K^{\mathfrak{S}}(\boldsymbol{\unicode[STIX]{x1D70E}})$ of totally $\mathfrak{S}$ -adic algebraic numbers that were introduced and studied by Geyer and Jarden and by Haran, Jarden, and 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 and Kiehne, Fried, Haran, and Völklein, and Ershov.