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

Truman

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 = 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 G´omez Ayala has determined a criterion for OL to be a free OK[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 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 G´omez Ayala’s criterion for the normal case.

• Acceptance Date: Sep 11, 2019
• Publication Date: May 1, 2020
• Journal: Journal of Pure and Applied Algebra
• Print ISSN: 0022-4049
• Publisher: Elsevier
• DOI: https://doi.org/10.1016/j.jpaa.2019.106231
• Keywords: Hopf-Galois structure, Hopf-Galois module theory, Galois; module structure, Associated order
• Publisher URL: https://doi.org/10.1016/j.jpaa.2019.106231