Commutative Hopf-Galois module structure of tame extensions

We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of p-adic fields or number fields which is H-Galois for a commutative Hopf algebra H. Firstly, we show that if L/K is a tame Gable extension of p-adic fields then each fractional ideal of L is free over its associated order in H. We also show that this conclusion remains valid if L/K is merely almost classically Galois. Finally, we show that if L/K is an abelian extension of number fields then every ambiguous fractional ideal of L is locally free over its associated order in H. (C) 2018 Elsevier Inc. All rights reserved