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

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.