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Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if 
$K$ is a finitely generated extension of transcendence degree 
$r$ of a global field and 
$A$ is a closed abelian subgroup of 
$\textrm{Gal}(K)$, then 
${\mathrm{rank}}(A)\le r+1$. Moreover, if 
$\mathrm{char}(K)=0$, then 
${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of 
$\textrm{Gal}(K)$.
$\mathfrak{S}$ -ADIC NUMBERS
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.
