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 pth root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G=Gal(L/K)) gives the unique Hopf-Galois structure on L/K, the ring of algebraic integers OL is locally free over OK[G] by Noether's theorem, and Gómez Ayala has determined a criterion for OL to be a free OK[G]-module. If K does not contain a primitive pth 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 OL 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 Gómez Ayala's criterion for the normal case.