On the monoid of tame extensions

This paper deals with not necessarily maximal orders in abelian extensions of number fields. We restrict our attention to orders invariant under the Galois group $G$. Based on recent work of Childs and Hurley [CH], we introduce a notion of tameness for such orders (actually this is done in a slightly more general setting). The maximal order is tame in this sense if and only if the field extension is tamely ramified.