Maximal tame extensions over Hopf orders in rings of integers of -adic number fields

Let K/k be a cyclic totally ramified Kummer extension of degree pn with Galois group G. Let and be the rings of integers in K and k, respectively. For n=1, F. Bertrandias and M.-J. Ferton determined the ring of -endomorphisms of and L.N. Childs constructed the maximal order S in whose ring of -endomorphisms is a Hopf order. A Hopf order whose linear dual is a Larson order in kG is called a dual Larson order. In this paper, the generators of a dual Larson order are given. We determine and obtain a maximal dual Larson order contained in . We construct a Hopf Galois extension S in , whose ring H(S) of -endomorphisms is a dual Larson order. We affirm an order S′ in with a dual Larson order H(S′) is a tame H(S′)-extension and show that S is maximal in such orders S′.