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 $\OL$ is free over its associated order $\AH$ 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.