On tamely ramified infinite Galois extensions

Abstract

For a number field $K$, we consider $K^{\mathrm{ta}}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G_K^{\mathrm{ta}}=Gal(K^{ta}/K)$. Choose a prime $p$ such that ${\mu }_p\not\subset K$. Our guiding aim is to characterize the finitely generated pro-$p$ quotients of~$G^{\mathrm{ta}}$. We give a {unified point of view} by introducing the notion of {\it stably inertially generated} pro-$p$ groups~$G$, for which linear groups are archetypes. This key notion {is compatible} with local {\it tame liftings} as used in the Scholz-Reichardt Theorem. We realize every finitely generated pro-$p$ group~$G$ which is stably inertially generated as a quotient of $G^{\mathrm{ta}}$. Further examples of groups that we realize as quotients of $G^{\mathrm{ta}}$ include congruence subgroups of special linear groups over ${\mathbb{Z}}_p[[T_1,\cdots ,T_n]]$. Finally, we give classes of groups which cannot be realized as quotients of $G_{\mathbb{Q}}^{\mathrm{ta}}$.