Towards a generalisation of Noether's theorem to nonclassical Hopf-Galois structures

We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of p-adic fields and number fields which are at most tamely ramified. We show that if L/K is an unramified extension of p-adic fields which is H-Galois for some Hopf algebra H then D-L is free over its associated order U-H in H. If H is commutative, we show that this conclusion remains valid in ramified extensions of p-adic fields if p does not divide the degree of the extension. By combining these results we prove a generalisation of Noether's theorem to nonclassical Hopf-Galois structures on domestic extensions of number fields.