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Let $ K $ be a number field and let $ L/K $ be a tamely ramified radicalextension of prime degree $ p $. If $ K $ contains a primitive $ p^{th} $ rootof unity then $ L/K $ is a cyclic Kummer extension; in this case the groupalgebra $ K[G] $ (with $ G=\mbox{Gal}(L/K) $) gives the unique Hopf-Galoisstructure on $ L/K $, the ring of algebraic integers $ \mathfrak{O}_L $ islocally free over $ \mathfrak{O}_{K}[G] $ by Noether's theorem, and G\'{o}mezAyala 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, whichagain admits a unique Hopf-Galois structure. Under the assumption that $ p $ isunramified in $ K $, we show that $ \mathfrak{O}_L $ is locally free over itsassociated order in this Hopf-Galois structure and determine a criterion for itto be free. We find that the conditions that appear in this criterion areidentical to those appearing in G\'{o}mez Ayala's criterion for the normalcase.

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