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

Truman, Paul J.

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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.

Journal Article Type Article
Acceptance Date Sep 11, 2019
Online Publication Date Sep 12, 2019
Publication Date 2020-05
Publicly Available Date Mar 29, 2024
Journal Journal of Pure and Applied Algebra
Print ISSN 0022-4049
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 224
Issue 5
Article Number 106231
DOI https://doi.org/10.1016/j.jpaa.2019.106231
Keywords Hopf-Galois structure, Hopf-Galois module theory, Galoismodule structure, Associated order
Publisher URL https://doi.org/10.1016/j.jpaa.2019.106231

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