The absolute Galois group of the field of totally $S$-adic numbers

Abstract

For a finite set $S$ of primes of a number field $K$ and for $\sigma_{1}, \dots, \sigma_{e} \in \operatorname{Gal}(K)$ we denote the field of totally $S$-adic numbers by $K_{{\rm tot}, S}$ and the fixed field of $\sigma_{1}, \dots, \sigma_{e}$ in $K_{{\rm tot}, S}$ by $K_{{\rm tot}, S}({\boldsymbol\sigma})$. We prove that for almost all ${\boldsymbol\sigma} \in \operatorname{Gal}(K)^{e}$ the absolute Galois group of $K_{{\rm tot}, S}({\boldsymbol\sigma})$ is the free product of ${\hat F}_{e}$ and a free product of local factors over $S$.