Maximal tame extensions over Hopf orders in rings of integers of [formula omitted]-adic number fields

Abstract

Let K/ k be a cyclic totally ramified Kummer extension of degree p n with Galois group G. Let O and o be the rings of integers in K and k, respectively. For n=1, F. Bertrandias and M.-J. Ferton determined the ring End oG ( O) of oG -endomorphisms of O and L.N. Childs constructed the maximal order S in O whose ring of oG -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 End oG ( O) and obtain a maximal dual Larson order contained in End oG ( O) . We construct a Hopf Galois extension S in O , whose ring H( S) of oG -endomorphisms is a dual Larson order. We affirm an order S′ in O with a dual Larson order H( S′) is a tame H( S′)-extension and show that S is maximal in such orders S′.