On tamely ramified infinite Galois extensions

Abstract For a number field , we consider the maximal tamely ramified algebraic extension of , and its Galois group . Choose an odd prime . Our guiding aim is to characterize the finitely generated pro‐ quotients of . We give a unified point of view by introducing the notion of stably inertially generated pro‐ groups , for which linear groups are archetypes. This key notion is compatible with local tame liftings as used in the Scholz–Reichardt theorem. We realize every finitely generated pro‐ group which is stably inertially generated as a quotient of . Further examples of groups that we realize as quotients of include congruence subgroups of special linear groups over . Finally, we give classes of groups which cannot be realized as quotients of .