# Hopf-Galois module structure of tamely ramified radical extensions of prime degree

## TL;DR

This paper investigates the module structure of tamely ramified radical extensions of prime degree over number fields, establishing conditions for the ring of integers to be free over associated orders in Hopf-Galois structures.

## Contribution

It extends the understanding of Hopf-Galois module structures to non-normal radical extensions, providing criteria for freeness similar to the normal case.

## Key findings

- The ring of integers is locally free over the associated order in the Hopf-Galois structure.
- Criteria for freeness are identical in both normal and non-normal cases under certain conditions.
- The extension's ramification and roots of unity influence the module structure and freeness conditions.

## Abstract

Let $ K $ be a number field and let $ L/K $ be a tamely ramified radical extension of prime degree $ p $. If $ K $ contains a primitive $ p^{th} $ root of unity then $ L/K $ is a cyclic Kummer extension; in this case the group algebra $ K[G] $ (with $ G=\mbox{Gal}(L/K) $) gives the unique Hopf-Galois structure on $ L/K $, the ring of algebraic integers $ \mathfrak{O}_L $ is locally free over $ \mathfrak{O}_{K}[G] $ by Noether's theorem, and G\'{o}mez Ayala has determined a criterion for $ \mathfrak{O}_L $ to be a free $ \mathfrak{O}_{K}[G] $-module. If $ K $ does not contain a primitive $ p^{th} $ root of unity then $ L/K $ is a separable, but non-normal, extension, which again admits a unique Hopf-Galois structure. Under the assumption that $ p $ is unramified in $ K $, we show that $ \mathfrak{O}_L $ is locally free over its associated order in this Hopf-Galois structure and determine a criterion for it to be free. We find that the conditions that appear in this criterion are identical to those appearing in G\'{o}mez Ayala's criterion for the normal case.