Towards a Generalisation of Noether's Theorem to Nonclassical Hopf-Galois Structures

TL;DR

This paper extends Noether's theorem to nonclassical Hopf-Galois structures in certain algebraic number field extensions, demonstrating conditions under which rings of integers are free modules over associated orders.

Contribution

It generalizes Noether's theorem to nonclassical Hopf-Galois structures in tame extensions, providing new conditions for module freeness.

Findings

Rings of integers are free over their associated orders in unramified $p$-adic extensions.

The freeness result extends to ramified extensions when the Hopf algebra is commutative and $p$ does not divide the extension degree.

A generalization of Noether's theorem is established for domestic extensions of number fields.

Abstract

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.